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I 


AMERICAN  MATHEMATICAL  SOCIETY 
COLLOQUIUM  LECTURES,  VOLUME  IV 


THE  MADISON  COLLOQUILM 

1913 

I.    ON  INVARIANTS  AND  THE  THEORY 

OF  NUMBERS 

BY 

LEONARD  EUGENE  DICKSON 


II.  TOPICS  IN  THE  THEORY  OF  FUNCTIONS 
OF  SEVERAL  COMPLEX  VARIABLES 


BY 

WILLIA:M  FOGG  OSGOOD 


NEW  YORK 

PUBLISHED  BY  THE 

AMERICAN  MATHEMATICAL  SOCIETY 

501  West  116th  Street 

1914 

3^7  5^11 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER    PA. 


PREFACE 


Sciencas 
Ubraiiy 


Shortly  after  its  reorganization  in  1894  as  a  national  body  the 
American  Mathematical  Society  inaugurated  the  plan  of  holding, 
at  intervals  of  two  to  four  years,  CoUoquia  or  courses  of  lectures 
given  by  representative  members  in  their  special  jBelds.  The 
Seventh  Colloquium  of  the  Society  was  held  at  Madison,  Septem- 
ber 10-13,  1913. 

Before  the  Society  became  a  national  organization,  a  Collo- 
quium was  held  at  Evanston,  in  1893,  at  which  Professor  Klein, 
of  Gottingen,  was  the  sole  speaker.*     Then  followed: 

The  Buffalo  Colloquhhm,  1896 

"(a)  Professor  Maxime  Bocher,  of  Harvard  University:  Linear 
Differential  Equations,  and  Their  Applications. 
This  Colloquium  has  not  been  published,  but  several  papers 
appeared  at  about  the  time  of  the  Colloquium,  in  which  the 
author  dealt  with  topics  treated  in  the  lectures. f 
(6)  Professor   James   Pierpont,   of  Yale   University:   Galois's 
Theory  of  Equations, 
Published  in  the  Annals  of  Mathematics,  series  2,  volumes  1 
and  2  (1900). 

The  Cambridge  Colloquium,  1898 
(a)  Professor   William   F.    Osgood,    of   Harvard   University: 
Selected  Topics  in  the  Theory  of  Functions. 
Published  in  the  Bulletin  of  the  American  Mathematical  Society, 
volume  5  (1898),  pages  59-87. 

*  The  first  edition  of  this  Colloquium  was  exhausted,  and  a  second^edition 
was  published  by  the  Society.  The  title  is:  The  Evanston  Colloquium, 
Lectures  on  Mathematics;  New  York,  American  Mathematical  Society,  1911. 

t  Two  of  these  papers  were:  "Regular  points  of  linear  differential  equations 
of  the  second  order,"  Harvard  University,  1896;  "Notes  on  some  points  in  the 
theory  of  linear  differential  equations,"  Annals  of  Mathematics,  vol.  12  (1S98). 

1 


11  PREFACE. 

(6)  Professor  Arthur  G.  Webster,  of  Clark  University:  The 
Partial  Differential  Equations  of  Wave  Propagation. 

The  Ithaca  Colloquium,  1901 

(a)  Professor  Oskar  Bolza,  of  the  University  of  Chicago:  The 

Simplest  Type  of  Problems  in  the  Calculus  of  Variations. 
Published  in  amplified  form  under  the  title:  Lectures  on  the 
Calculus  of  Variations,  Chicago,  1904, 

(b)  Professor  Ernest  W.  Brown,  of  Haverford  College:  Modern 

Methods  of  Treating  Dynamical  Problems,  and  in  Par- 
ticular the  Problem  of  Three  Bodies. 

Beginning  with  the  lectures  of  1903,  the  Colloquia  have  been 
published  as  monographs,  and  are  here  numbered  accordingly. 

I.    The  Boston  Colloquium,  1903 

(a)  Professor  Henry  S.  White,  of  Northwestern  University: 

three  lectures  on  Linear  Systems  of  Curves  on  Algebraic 
Surfaces. 

(b)  Professor  Frederick  S.  Woods,  of  the  Massachusetts  In- 

stitute of  Technology:  three  lectures  on  Forms  of  Non- 
Euclidean  Space. 

(c)  Professor  Edward  B.  Van  Vleck,  of  Wesleyan  University: 

six  lectures  on  Selected  Topics  in  the  Theory  of  Divergent 
Series  and  Continued  Fractions. 
Published  for  the  Society  under  the  title:  The  Boston  Collo- 
quium Lectures  on  Mathematics.     New  York,  The  Macmillan 
Company,  1905. 

II.    The  New  Haven  Colloquium,  1906 

(a)  Professor  Eliakim  H.  Moore,  of  the  University  of  Chicago: 

five  lectures  on  an  Introduction  to  a  Form  of  General 
Analysis. 

(b)  Professor  Ernest  J.  Wilczynski,  of  the  University  of  Cali- 

fornia: four  lectures  on  Projective  Differential  Geometry. 


PREFACE.  ill 

(c)  Professor  Max  Mason,  of  Yale  University:  four  lectures  on 
Selected  Topics  in  the  Theory  of  Boundary  Value  Problems 
of  Differential  Equations. 
Published   under  the  title:   The   New  Haven  Mathematical 

Colloquium.     Yale  University  Press,  1910. 

III.     The  Princeton  Colloquium,  1909 
(a)  Professor  Gilbert  A.  Bliss,  of  the  University  of  Chicago: 

four  lectures  on  Fundamental  Existence  Theorems. 

(6)  Professor  Edward  Kasner,  of  Columbia  University:  four 

lectures  on  Differential-Geometric  Aspects  of  Dynamics. 

Published  under  the  title :  The  Princeton  Colloquium  Lectures 

on  Mathematics.     New  York,  American  ^Mathematical  Society, 

1913. 

It  is  contemplated  that  the  Society  will  henceforth  regularly 
publish  the  Colloquia,  and  thus  the  present  volume  appears  as 
Volume  IV  in  the  series. 

IV.    The  Madison  Colloquium,  1913 

(a)  Professor  Leonard  E.  Dickson,  of  the  University  of  Chicago : 

five  lectures  on  Invariants  and  the  Theory  of  Numbers. 

(b)  Professor  William  F.  Osgood,  of  Harvard  University:  five 

lectures  on  Topics  in  the  Theory  of  Functions  of  Several 
Complex  Variables. 


ON  INVARIANTS  AND  THE 
THEORY  OF  NUMBERS 


BY 


LEONARD  EUGENE  DICKSON 


CONTENTS 


Page 

Introduction 1 


LECTURE    I 

A   Theory   of   Invariants   Applicable   to  Algebraic  and 

Modular  Forms 

1-3.  Introduction  to  the  algebraic  side  of  the  theory  by 
means  of  the  example  of  an  algebraic  quadratic 

form  in  m  variables 4 

4-7.  Introduction  to  the  number  theory  side  of  the 
theory  of  invariants  by  means  of  the  example  of 

a  modular  quadratic  form 6 

8-9.     Modular  invariants  are  rational  and  integral ......  12 

10.  Characteristic  modular  invariants 13 

11.  Number  of  linearly  independent  modular  invariants  13 

12.  Fundamental  system  of  modular  invariants 14 

13.  Minor  role  of  modular  covariants 15 

14.  References  to  further  developments 15 

LECTURE   II 

Seminvariants  of  Algebraic  and  Modular  Binary  Forms 

1-6.     Introductory  example  of  the  binary  quartic  form.  .     16 
7-10.     Fundamental  system  of  modular  seminvariants  of  a 

binary  n-ic  derived  by  induction  from  n—  \to  n    21 
11.     Explicit  fundamental  system  when  the  modulus  y 

exceeds  n 25 


11  CONTENTS. 


12.  Another  method  for  the  case  p  >  n 27 

13.  Number  of  linearly  independent  seminvariants ...  .  28 
14-15.     Derivation    of    modular    invariants    from    semin- 
variants    28 


LECTURE   III 
Inyaeiants  of  a  Modular  Group.    Formal  Invariants  and 

COVARL^NTS   of   JNIoDULAR   FoRMS.      APPLICATIONS 

1-4.     Invariants  of  certain  modular  groups;  problem  of 

Hurwitz 33 

5-11.     Formal    invariants    and    seminvariants    of   binary 

modular  forms 40 

12.  Theorem  of  Miss  Sanderson 54 

13.  Fundamental  systems  of  modular  CO  variants 55 

14.  Form  problem  for  the  total  binary  modular  group  58 

15.  Invariantive  classification  of  forms 61 


LECTURE  IV 

Modular  Geometry  and  Covariantive  Theory  of  a  Quad- 
ratic Form  in  m  Variables  INIodulo  2 

1-2.     Introduction.     The  polar  locus 65 

3.  Odd  number  of  variables;  apex;  linear  tangential 

equation 66 

4.  Covariant  line  of  a  conic 69 

5.  Even  number  of  variables 70 

6.  Covariant  plane  of  a  degenerate  quadric  surface  .  .  71 

7.  A  configuration  defined  by  the  quinary  surface ....  72 

8.  Certain  formal  and  modular  covariants  of  a  conic  73 
9-32.     Fundamental  system  of  covariants  of  a  conic 76 

33.     References  on  modular  geometry 98 


CONTENTS.  Ill 

LECTURE  V 

A  Theory  of  Plane  Cubic  Curves  with  a  Real  Inflexion 
Point  Valid  in  Ordinary  and  in  Modular  Geometry 

1.  Normal  form  of  a  ternary  cubic 99 

2.  The  invariants  s  and  t 99 

3-  The  four  inflexion  triangles 100 

4.  The  parameter  5  in  the  normal  form 101 

5-9.     Criteria  for  9, 3  or  1  real  inflexion  points;  sub-cases  101 


ON  INVARIANTS  AND  THE  THEORY 
OF  NUMBERS 

BY 

LEONARD  EUGENE  DICKSON 


INTRODUCTION 

A  simple  theory  of  invariants  for  the  modular  forms  and  linear 
transformations  employed  in  the  theory  of  numbers  should  be 
of  an  importance  commensurate  with  that  of  the  theory  of 
invariants  in  modern  algebra  and  analytic  projective  geometry, 
and  should  have  the  advantage  of  introducing  into  the  theory 
of  numbers  methods  uniform  with  those  of  algebra  and  geometry. 

In  considering  the  invariants  of  a  modular  form  (a  homo- 
geneous polynomial  with  integral  coefficients  taken  modulo  p, 
where  p  is  a  prime),  we  see  at  once  that  the  rational  integral 
invariants  of  the  corresponding  algebraic  form  with  arbitrary 
variables  as  coefficients  give  rise  to  as  many  modular  invariants 
of  the  modular  form,  and  that  there  are  numerous  additional 
invariants  peculiar  to  the  case  of  the  theory  of  numbers.  More- 
over, nearly  all  of  the  processes  of  the  theory  of  algebraic  in- 
variants, whether  symbolic  or  not,  either  fail  for  modular  in- 
variants or  else  become  so  complicated  as  to  be  useless.  For 
instance,  the  annihilators  are  no  longer  linear  differential  oper- 
ators. The  attempt  to  construct  a  simple  theory  of  modular 
invariants  from  the  standpoints  in  vogue  in  the  algebraic  theory 
was  a  failure,  although  useful  special  results  were  obtained  in 
this  laborious  way.  Later  I  discovered  a  new  standpoint  which 
led  to  a  remarkably  simple  theory  of  modular  invariants.  This 
standpoint  is  of  function-theoretic  character,  employing  the 
2  1 


2  THE  MADISON   COLLOQUIUM. 

values  of  the  invariant,  and  using  linear  transformations  only  in 
the  preliminary  problem  of  separating  into  classes  the  particular 
forms  obtained  by  assigning  special  values  to  the  coefficients  of 
the  ground  form.  As  to  the  practical  value  of  the  new  theory  as 
a  working  tool,  it  may  be  observed  that  the  problem  to  find  a 
fundamental  system  of  modular  seminvariants  of  a  binary  form  is 
from  the  new  standpoint  a  much  simpler  problem  than  the  cor- 
responding one  in  the  algebraic  case;  indeed,  we  shall  exhibit 
explicitly  the  fundamental  system  of  modular  seminvariants  for  a 
binary  form  of  general  degree. 

It  will  now  be  clear  why  these  Lectures  make  no  use  of  the 
technical  theories  of  algebraic  invariants.  On  the  contrary,  they 
afford  an  introduction  to  that  subject  from  a  new  standpoint 
and,  in  particular,  throw  considerable  new  light  on  the  relations 
between  the  subjects  of  rational  integral  invariants  and  tran- 
scendental invariants  of  algebraic  forms  and  the  corresponding 
questions  for  seminvariants.  Again,  I  shall  make  no  use  of 
technical  theory  of  numbers,  presupposing  merely  the  concepts 
of  congruence  and  primitive  roots,  Fermat's  theorem,  and  (in 
Lectures  III  and  V)  the  concept  of  quadratic  residues. 

The  developments  given  in  these  Lectures  are  new,  with 
exceptions  in  the  case  of  Lecture  I,  which  presents  an  intro- 
duction to  the  theory,  and  in  the  case  of  the  earlier  and  final 
sections  of  Lecture  III.  But  in  these  cases  the  exposition  is 
considerably  simpler  and  more  elementary  than  that  in  my 
published  papers  on  the  same  topics.  The  contacts  with  the 
work  of  other  writers  will  be  indicated  at  the  appropriate  places. 
Much  light  is  thrown  upon  the  unsolved  problem  of  Hurwitz 
concerning  formal  invariants. 

In  many  parts  of  these  Lectures,  I  have  not  aimed  at  complete 
generality  and  exhaustiveness,  but  rather  at  an  illumination  of 
typical  and  suggestive  topics,  treated  by  that  particular  method 
which  I  have  found  to  be  the  best  of  various  possible  methods. 
Surely  in  a  new  subject  in  which  most  of  the  possible  methods  are 
very  complex,  it  is  desirable  to  put  on  record  an  account  of  the 


INVARIANTS  AND  NUMBER  THEORY.  3 

simple  successful  methods.  Finally,  it  may  be  remarked  that 
the  present  theory  is  equally  simple  when  the  coefficients  of  the 
forms  and  linear  transformations  are  not  integers,  but  are  ele- 
ments of  any  finite  field. 

I  am  much  indebted  to  Dr.  Sanderson  and  Professors  Cole 
and  Glenn  for  reading  the  proof  sheets. 


LECTURE  I 

A  THEORY  OF  INVARIANTS  APPLICABLE  TO  ALGEBRAIC  AND 

MODULAR   FORMS 

Introduction  to  the  Algebraic  Side  of  the  Theory  by 

Means  of  the  Example  of  an  Algebraic  Quadratic 

Form  in  m  Variables,  §§  1-3 

1.  Classes  of  Algebraic  Quadratic  Forvis. — Let  the  coefficients  of 

m 
(1)  gm  =    2  ^ijXiXj  (J^ji  =  fiij) 

be  ordinary  real  or  complex  numbers.     Let  the  determinant 

(2)  D=  \l3ij\  (i,j=  1,  '",m) 

of  a  particular  form  qm  be  of  rank  r  (r  >  0);  then  every  minor  of 
order  exceeding  r  is  zero,  while  at  least  one  minor  of  order  r  is 
not  zero.  There  exists  a  linear  transformation  of  determinant 
unity  which  replaces  this  q^  by  a  form* 

(3)  ai^Ti^  H h  ar.Tr-      (ai  +  0,  .  •  • ,  a.  +  0). 

Indeed,  if  fin  4=  0,  we  obtain  a  form  lacking  .Ti.ro,  •  •  • ,  XiXm  by 
substituting 

xi  —  fin~KfinX2  +  •  •  •  +  fiiviXm) 

for  .Ti.  If  fin  =  0,  fin  =r  0,  we  substitute  Xi  for  Xi  and  —  .ri 
for  .Ti;  while,  if  every  fikk  =  0,  and  fin  4=  0,  we  substitute 
X2  +  Xi  for  X2;  in  either  case  we  obtain  a  form  in  which  the  co- 
eflficient  of  x{-  is  not  zero.  We  now  have  ai.Ti^  +  (f>,  where 
ai  4=  0  and  0  involves  only  X2,  •  •  • ,  Xm.  Proceeding  similarly 
with  0,  we  ultimately  obtain  a  form  (3). 

Now  (3)  is  replaced  by  a  similar  form  having  ai  =  1  by  the 

*  Note  for  later  use  that  each  ak  and  each  coefficient  of  the  transformation 
is  a  rational  function  of  the  /3's  with  integral  coefficients. 

4 


INVARIANTS   AND   NUMBER  THEORY.  5 

transformation 

.Ti  =  arhi,     .r„,  =  diKi'm',     Xi  =  x/     (i  =  2,  •  •  •,  7?i  —  1) 

of  determinant  unity.  Hence  there  exists  a  linear  transforma- 
tion with  complex  coefficients  of  determinant  unity  which 
replaces  qm  by 

(4)  .Ti^  +  .  •  •  +  xl-i  +  DxJ,        X,'  +  •  •  •  +  Xr', 

according  as  r  =  m  or  r  <  m.  In  the  first  case,  the  final  co- 
efficient is  D  since  the  determinant  (2)  of  a  form  qm  equals  that 
of  the  form  derived  from  qm  by  any  linear  transformation  of  de- 
terminant unity.  Hence  all  quadratic  forms  (1)  may  be  separated 
into  the  classes 

(5)  C^.B,     Cr     (Z)  4=0w  =  0,  1,  •••,m-l), 

where,  for  a  particular  number  Z)  4=  0,  the  class  C^,  i,  is  composed 
of  all  forms  qm  of  determinant  D,  each  being  transformable  into 
(4i) ;  while,  for  0  <  r  <  m,  the  class  Cr  is  composed  of  all  forms 
of  rank  r,  each  being  transformable  into  (42);  and,  finally,  the 
class  Co  is  composed  of  the  single  form  with  every  coefficient 
zero.  In  the  last  case,  the  determinant  D  is  said  to  be  of  rank 
zero.  Using  also  the  fact  that  the  rank  of  the  determinant  of  a 
quadratic  form  is  not  altered  by  linear  transformation,  we  con- 
clude that  two  quadratic  fonns  are  tramformahle  into  each  other 
by  linear  transformations  of  determinant  unity  if  and  only  if  they 
belong  to  the  same  class  (5). 

2.  Single-valued  Invariants  of  q^. — Using  the  term  function 
in  Dirichlet's  sense  of  correspondence,  we  shall  say  that  a  single- 
valued  function  0  of  the  undetermined  coefficients  /3,7  of  the 
general  quadratic  form  q^,  is  an  invariant  of  qm  if  </>  has  the  same 
value  for  all  sets  /3^,  jS^^.,  •  •  •  of  coefficients  of  forms  q',„  ql,  •  •  • 
belonging  to  the  same  class.*  The  values  i'„,,  o,  iv  of  (f>  for  the 
various  classes  (5)  are  in  general  different.  For  example,  the 
determinant  D  is  an  invariant;  likewise  the  single- valued  func- 

*  Briefly,  if  <j>  has  the  same  value  for  all  forms  in  any  class. 


6  THE   MADISON   COLLOQUIUM. 

tion  r  of  the  undetermined  coefficients  ^a  which  specifies  the  rank 
of  liSiil- 

Each  consistent  set  of  values  of  D  and  r  uniquely  determines  a 
class  (5)  and,  by  definition,  each  class  uniquely  determines  a 
value  of  0.     Hence  0  is  a  single  valued  function  of  D  and  r. 

Every  single-valued  invariant  of  a  system  of  forms  is  a  single- 
valued  function  of  the  invariants  {D  and  r  in  our  example)  which 
completely  characterize  the  classes. 

3.  Rational  Integral  Invariants  of  qm. — If  the  invariant  <^  is  a 
rational  integral  function  of  the  coefficients  fin,  it  equals  a  rational 
integral  function  of  D.  For,  if  the  |S's  have  any  values  such 
that  i)  =t=  0,  0  has  the  same  value  for  the  form  (1)  as  for  the 
particular  form  (4i)  of  the  same  class.  Hence  <i>  =  P{D), 
where  P(-D)  is  a  polynomial  in  D  with  numerical  coefficients. 
Since  this  equation  holds  for  all  sets  of  jS's  whose  determinant 
is  not  zero,  it  is  an  identity. 

Introduction  to  the  Number  Theory  Side   of  the 

Theory  of  Invariants  by  Means  of  the  Example 

OF  A  Modular  Quadratic  Form,  §§  4-7 

4.  Classes  of  Modular  Quadratic  Forms  qm. — Let  Xi,  •  •  • ,  Xm  be 
indeterminates  in  the  sense  of  Kronecker.  Let  each  fiij  be  an 
integer  taken  modulo  p,  where  p  is  an  odd  prime.  Then  the 
expression  (1)  is  called  a  modular  quadratic  form.  By  §  1,  there 
exists  a  linear  transformation,  whose  coefficients  are  integers* 
taken  modulo  p  and  whose  determinant  is  congruent  to  unity, 
which  replaces  qm  by  a  quadratic  form  (3)  in  which  each  ak  is  an 
integer  not  divisible  by  p.  Thusf  each  ak  is  congruent  to  a 
power  of  a  primitive  root  p  of  p.  After  applying  a  linear  trans- 
formation of  determinant  unity  which  permutes  Xi^,  •  •  • ,  Xr^, 
we  may  assume  that  ai,  •  •  •,  ««  are  even  powers  of  p  and  that 
aa+i,  •  •  • ,  ar  are  odd  powers  of  p.     The  transformation  which 

*  Perhaps  initially  of  the  form  a/b,  where  a  and  b  are  integers,  b  not  divisible 
by  p.     But  there  exists  an  integral  solution  q  oi  qb  =  a  (mod  p). 
t  For  p  =  5,  p  =  2,  1  s  2",  2  =  2^,  3  =  2»,  4  =  2^  (mod  5). 


INVARIANTS  AND  NUMBER  THEORY.  7 

multiplies  a  particular  Xi  {i  <  m)  by  p^  and  Xm  by  p~^  is  of  de- 
terminant unity. 

First,  let  r  <  m.  Applying  transformations  of  the  last  type 
to  (3),  we  obtain 

(6)  a:i2  +  . . .  +  xs^  +  pxl^,  +  •  •  •  +  pxr\ 
Under  the  transformation  of  determinant  unity 

Xi  =  aXi  +  PXj,      .Ty  =    -  pXi  +  aXj,      Xm  =   («2  +  ^T'Xm, 

X?  +  x^  becomes  {o?  +  ^"){X.^  +  ^/).  Choose*  integers  a,  /3 
so  that 

(7)  p(a2  +  /32)  =  1         (mod  2?). 

Hence  the  sum  of  two  terms  of  (6)  with  the  coefficient  p  can  be 
transformed  into  a  sum  of  two  squares.  Thus  by  means  of  a 
linear  transformation,  with  integral  coefficients  of  determinant 
unity,  Qm  can  be  reduced  to  one  of  the  forms 

(8)  .Ti2  +  . . .  +  xU,  +  xr\     xi'  +  •  •  •  +xU+pXr'     (0<r<m). 
Next,  let  r  =  m.     We  obtain  initially 

xi^  -{-•'-  -\-  x,^  -{-  px]+^  +  •  •  •  +  P-^l-i  +  <^xj, 

in  which  cr  need  not  equal  p  as  in  (6).  If  there  be  an  even  number 
of  terms  with  the  coefficient  p,  we  obtain  as  above  a  form  of 
type  (4i).     In  the  contrary  case,  we  get 

/  =  a:i2  +  • .  •  H-  xl_^  +  pxl_^  +  p-^DxJ. 

If  D  =  p^'+^  (mod  p),f  is  transformed  into  (40  by 

Xm—l  ^^  P  -'imj       Xni  =   p      A.^—!'  i/ 

But  if  Z)  =  p2',  /  is  reduced  to  (4i)  by  the  transformation 

Xrr^X  =   OiXm-l  +   5p-'~^A',„,       Xyn  =    —   8Xni-\  +  CXpXm, 

P(ci'  +  P^'-^5^)  ^1,  iX 

*  If  p  =  5,  p  =  2,  we  may  take  a  =  /3  =2.  For  any  p,  either  there  is  an 
integer  I  such  that  P  =  —  I  (mod  p)  and  we  may  take  pia  +  13)  =  1, 
a  —  Z/3  =  1;  or  else  x^  +  1  takes  1  +  (p  —  l)/2  incongruent  values  modulo  p, 
no  one  divisible  by  p,  when  a;  ranges  over  the  integers  0,  1,  •  •  •,  p  —  1,  so 
that  a;2  +  1  takes  at  least  one  value  of  the  form  p^*"'.  In  the  latter  event, 
a.  =  p~*,  0  =  xa  satisfy  (7). 


8  THE   MADISON   COLLOQUIUM. 

of  determinant  unity.     The  final  condition  is  of  the  form    (7) 
with  /3  =  p'~^5  and  hence  has  integral  solutions  a,  5. 
Hence  the  classes  of  modular  quadratic  forms  are 

(9) 

(Z)  =  1,  •••,  p  -  1;  r  =  1,  '■-,  m—  1), 

where  Cm,  d  'is>  composed  of  all  modular  quadratic  forms  whose 
determinant  is  a  given  integer  D  not  divisible  by  p,  each  being 
transformable  into  (4i),  where  Cr,  i  and  Cr,  -i  are  composed  of 
all  forms  transformable  into  (81)  and  (82)  respectively,  and  Co  is 
composed  of  the  form  all  of  whose  coefficients  are  zero. 

Two  modular  quadratic  forms  are  transformable  into  each  other 
by  linear  transformations  ivith  integral  coefficients  of  determinant 
unity  modulo  p  if  and  only  if  they  belong  to  the  same  class  (9). 
Indeed,  since  D  and  r  are  invariants,*  it  remains  only  to  show 
that  the  two  forms  (8)  are  not  transformable  into  each  other. f 
But  if  a  linear  transformation 


Xi  =  ^aijXj                   (i=  1,  •  •  •,  m) 

replaces  /  =  .rr  +  • 

•  •  +  xr'  by  F=  X,'+--'  +  ZLi  +  pXr', 

then,  for  j  >  r, 

dXj      ~  j=i    *  dXj 

dX  ~  ^'     dX  ~  "''  ~  ^        (^  Sr,j>  r), 

r 

Xi  =  ^oiijXj                    (i  =  1,  •  •  •,  r). 

Hence  under  this  partial  transformation  on  Xi,  •  •  • ,  Xr,  we  would 
have  f  =  F.  Thus  the  determinant  of  F  would  equal  \<Xij\- 
times  the  determinant  unity  of  /  and  hence  equal  an  even  power 
of  p.     But  the  determinant  of  F  is  actually  p. 

*  r  is  now  the  maximum  order  of  a  minor  not  divisible  by  p. 

t  An  immediate  proof  follows  from  the  values  taken  by  the  invariant  Ar 
given  below.  But  as  the  necessity  of  constructing  Ar  is  based  upon  the  fact 
that  the  forms  (8)  do  not  belong  to  the  same  class,  it  seems  preferable  to  prove 
the  last  fact  without  the  use  of  Ar. 


INVAEIANTS   AND   NUMBER   THEORY.  9 

The  invariants  D  and  r  therefore  do  not  completely  characterize 
the  classes  of  modular  quadratic  forms,  a  result  in  contrast  to 
that  for  algebraic  quadratic  forms.  We  shall  give  a  criterion 
to  decide  whether  a  given  form  of  rank  r  (0  <  r  <  m)  is  of  class 
Cr,  1  or  of  class  Cr,  -i  and  later  deduce  an  invariantive  criterion. 

5.  Criterion  for  Classes  Cr,  ±i. — Such  a  criterion  may  be 
obtained  from  Kronecker's  elegant  theory  of  quadratic  forms.* 
We  shall  make  use  of  the  theorem  that  a  symmetrical  determinant 
of  rank  r  (r  >  0)  has  a  non-vanishing  principal  minor  M  of  order 
)•,  i.  e.,  one  whose  diagonal  elements  lie  in  the  main  diagonal  of 
the  given  determinant.!  After  an  evident  linear  transformation 
of  determinant  unity,  we  may  set 

(10)  M  =  \^ij\  ^  0         (mod  p)         (i,j=l,  ...,  ,-). 

In  the  present  problem,  /•  <  m.  To  g^  apply  the  transforma- 
tion 

Xi  =  Xi  +  dXm  (i  =  1,  •  •  •,  r), 

Xi  =  Xi  (i  =  r  +  1,  •  •  •,  /?0 

of  determinant  unity  in  which  the  c,-  are  integers.     We  get 

m— 1  '/«— 1  /     /•  \ 

2^   ^ijXiXj  +  2  2_/  BjmXjXm  -\-  [^  BjmCj  +  Bmm    I  Xm', 
i,  j=l  J=l  \  J=i  / 

where 

r 
Bjm  =   S  fiijCi  +  ^jm  U   =    I,    •••,    m). 

In  view  of  (10)  there  are  unique  values  of  Ci,  •  •  • ,  Cr  such  that 

Bjm=0         (mod  2>)  (i  =  1,  ••  •, /•)• 

But  the  determinant  of  the  coefficients  of  ci,  •  •  •,  Cr,  1  in 

Bim,     Bim,     •  •  • ,     Brm,     Bkm         (r  <  k  ^  m) 


♦Kronecker,  Werke,  vol.  1,  p.  166,  p.  357;  cf.  Gundelfinger,  Crelle,  vol.  91 
(1881),  p.  221;  Bocher,  Introduction  to  Higher  Algebra,  p.  58,  p.  139. 

tThe  most  elementary  proof  is  that  by  Dickson,  Annals  of  Mathcmalics, 
ser.  2,  vol.  15  (1913),  pp.  27,  28.  For  other  short  proofs,  see  Wedderburn, 
ibid.,  p.  29,  and  Kowalewski,  Determinantentheorie,  pp.  122-124. 


10  THE   MADISON   COLLOQUIUM. 

is  the  minor  of  ^km  in  the  determinant 

fe-|  (hj=  1,  "',r,k,m) 

and  hence  is  zero,  being  of  order  r  +  1.  Hence  Bkm  —  0.  Thus 
Qm  has  been  transformed  into 

m—l 

After  repetitions  of  this  process,  Qm  is  transformed  into* 

r 

(11)  S  fiijXiXj. 

This  form,  of  determinant  M,  can  be  reduced  (§  4)  to 

by  a  linear  transformation  on  a^i,  •  •  • ,  Xr  with  integral  coefficients 
of  determinant  unity  modulo  p.  Express  M  as  a  power  p^'^* 
(e  =  0  or  1)  of  a  primitive  root.  Since  r  <  m,  we  may  replace 
Xr  by  p~^Xt  and  Xm.  by  p'.T;„  and  obtain  (8i)  or  (82)  according  as 
€  =  0  or  €  =  1.  Now  p(p~^)/2  jg  j^Q^  congruent  to  unity,  but  its 
square  is  congruent  to  unity  modulo  p,  by  Fermat's  theorem; 
hence  it  is  =  —  1.     Thus,  in  the  respective  cases, 

(12)  M  2   =  +  1     or     -  1         (mod  p). 

Hence  if  a  form  is  of  rank  r  and  if  M  is  any  chosen  r-rowed 
principal  minor  not  divisible  by  p,  the  form  is  of  class  Cr,  1  or 
Cr,  -1  according  as  the  first  or  second  alternative  (12)  holds. 

6.  Invariantive  Criterion  for  Classes  Cr,  ±1. — A  function  which 
has  the  value  +  1  for  any  form  of  class  Cr,  +1,  the  value  —  1  for 
any  form  of  class  Cr.  -1,  and  the  value  zero  for  the  remaining 
classes  Cm,  d,  Co,  Ck,  ±1  {k  ^  r),  is  an  invariant  (§  2).  This 
functionf  is 

*  This  proof  and  the  results  in  §§  4-13  are  due  to  Dickson,  Transactions  of 
the  American  Mathematical  Society,  vol.  10  (1909),  pp.  123-133. 
t  Constructed  synthetically  in  the  paper  last  cited. 


INVAKIANTS  AND  NUMBER  THEORY.  U 

Ar  =  {Mi~  +  M2~  (1  -  il/i^i)  +  •  •  • 
(13) 

+  Mr^a  -  1/iM  •••(!-  m:z\)}u{i  -  r/'-i), 

where  Mi,  •  •  • ,  Mn  denote  the  principal  minors  of  order  r  taken 
in  any  sequence,  and  d  ranges  over  the  principal  minors  of  orders 
exceeding  r.  For,  if  any  d  ^  0,  the  rank  exceeds  r  and  ^^  =  0 
by  Fermat's  theorem.  Next,  let  every  d  ^  0,  so  that  the  rank 
is  r  or  less,  and  the  final  factor  in  (13)  is  congruent  to  unity. 
Then,  if  every  Mi  =  0,  the  rank  is  less  than  r  and  Ar^  0. 
But,  if  Ml  ^  0, 

Ar  =  Ml  -  ^  ±  1         (mod  2>)> 

by  (12),  the  sign  being  the  same  as  in  Cr,  ±i.     If  Mi  =  0,  M^  ^  0, 

p-i 
^,  =  3/2  2  =  ±  1         (mod  jy), 

etc.     Note  for  later  use  that 

(14)  Am  =  D^. 

7.  Rational  Integral  Invariants  of  q,n. — The  function 

(15)  7o  =  n(l  -  /3^7')      (^,  i  =  1,  ■ . . ,  m;  i  ^  j) 

has  the  value  1  for  the  form  (of  class  Co)  all  of  whose  coefficients 
are  zero  and  the  value  0  for  all  remaining  forms  qm,  and  hence 
is  an  invariant  of  qm-     We  now  have  rational  integral  invariants 

(16)  D,     Ai,     •••,    Am-u    h 

which  completely  characterize  the  classes  (9).  Hence,  by  the 
general  theorem  in  §  12,  any  rational  integral  invariant  of  the 
modular  form  qm  is  a  rational  integral  function  of  the  invariants 
(16)  with  integral  coefficients.  In  other  words,  invariants  (16) 
form  a  fundamental  system  of  rational  integral  invariants  of  qm- 
If  we  employ  not  merely,  as  before,  linear  transformations 
with  integral  coefficients  of  determinant  unity  modulo  y,  but 
those  of  all  determinants,  we  obtain  at  once  the  classes 

Cr,^x,     Co  (r  =  1,  •••,  m), 


i-m 


'm 


12  THE   MADISOX    COLLOQUItHM. 

and  see  that  these  are  characterized  by  Ai,  •••,  Am,  h-  The 
latter  therefore  form  a  fundamental  sj^stem  of  rational  integral 
absolute  invariants.  *  But  Z>  is  a  relative  invariant. 

General  Theory  of  Modular  Invarl\nts,  §§  8-14 

8.  Definitions. — Let  S  be  any  system  of  forms  in  a:i,  •  •  • ,  x„ 
wdth  undetermined  integral  coefficients  taken  modulo  p,  a  prime. 
Let  G  be  any  group  of  linear  transformations  on  xi,  •  ■  ■ ,  x, 
with  integral  coefficients  taken  modulo  p.  The  particular  systems 
S',  S",  •  •  • ,  obtained  from  S  hy  assigning  to  the  coefficients 
particular  sets  of  integral  values  modulo  y,  may  be  separated  into 
classes  Co,  Ci,  ■  •  • ,  C„_i  such  that  two  systems  belong  to  the 
same  class  if  and  only  if  they  are  transformable  into  each  other 
by  transformations  of  G. 

A  single-valued  function  0  of  the  coefficients  of  the  forms  in 
the  system  S  is  called  an  invariant  of  S  under  G  if,  for  z  =  0,  1, 
•  •  • ,  n  —  1,  the  function  0  has  the  same  value  Vi  for  all  systems 
of  forms  in  the  class  Ci.  ^ 

In  case  the  values  taken  by  0  are  integers  which  may  be  N 
reduced  at  will  modulo  j)  and  cpngruent  values  be  identified, 
the  invariant  is  called  modular.  Since  this  reduction  can  be 
effected  on  each  coefficient  of  the  modular  forms  comprising  our 
system  S,  any  rational  integral  invariant  of  S  is  a  modular 
invariant. 

An  example  of  a  non-modular  invariant  is  the  transcendental 
function  r  defining  the  rank  of  the  determinant  of  the  modular 
quadratic  form  g^.  The  values  of  r  are  evidently  not  to  be 
identified  when  merely  congruent  modulo  p.  However,  the 
residue  of  r  modulo  y  is  a  modular  invariant,  since 

(17)  r  =  A,-  +  2A.}  +  •  •  •  +  mAJ^         (mod  p). 

9.  Modular  Invariants  are  Rational  and  Integral. — Any  modular 
invariant  0  of  a  system  S  of  modular  forms  can  be  identified  with  a 
rational  integraL  function  (with  integral  coefficients)  of  the 
coefficients  Ci,  •  •  • ,  Cg  appearing  in  the  forms  of  the  system  S. 


INVARIANTS   AND   NUMBER  THEORY.  13 

For,  if 

<l>  =  '^ei, ....  e.    when     ci^  Ci,  '",  Cs  =  Cs         (mod  p), 
then  (f>  is  identically  congruent  (as  to  Ci,  •  •  • ,  Cs)  to 

p  —  J  s 

(18)  Z         re,....,e^Il{l-(C;-ei)^'], 

as  shown  by  Fermat's  theorem. 

10.  Characteristic  Modular  Invariants. — The  characteristic  in- 
variant 7^-  of  the  class  Ck  is  defined  to  be  that  modular  inva- 
riant which  has  the  value  unity  for  systems  of  forms  of  the 
class  Ck  and  the  value  zero  for  any  of  the  remaining  classes. 

For  example,  for  a  single  quadratic  form  qm,  Iq  is  given  by 
(15),' while  the  characteristic  invariants  for  the  classes  Cr,  i  and 
Cr,  -1  are 

(19)  Ir,l-hUr--\-  Ar),       I r,  -1   =   i{A/  -  Ar) .    ^ 

For  any  system  of  forms  with  the  coefficients  ci,  •  •  • ,  (?«,  we 
have 

(20)  h-Zllil-  (c.-c,^^>)^M, 

i=l 

where  the  sum  extends  over  all  sets  of  coefficients  Ci^''\  -  •  ■ ,  c^^^^ 
of  the  various  systems  of  forms  of  class  Ck-  In  particular,  in 
accord  with  (15), 

(21)  /o  =  n  (1  -  cr')- 

1=1 

11.  Number  of  Linearly  Independent  Modular  Invariants. — 
Since  aiiy  modular  invariant  /  takes  certain  values  Vq,  •  •  • ,  i'„-i 
for  the  respective  classes  Co,  •  •  • ,  Cn~i,  we  have 

(22)  I  =    Voh  +   l\Il  + h  Vn-lln-l. 

Hence  any  modular  invariant  can  be  expressed  in  one  and  but 
one  way  as  a  linear  homogeneous  function  of  the  characteristic 
invariants.  Moreover,  the  number  of  linearly  independent 
modular  invariants  equals  the  number  of  classes. 


14  THE   MADISON  COLLOQUIUM. 

For  example,  using  (19),  we  see  that  a  complete  set  of  linearly 
independent  modular  invariants  of  the  quadratic  form  q^ 
modulo  p  {p  >  2)  is  given  by 

(23)    7o,  Ar,  Ar'    ir=l,'--,m-  1),      D^    (k=l,  ...,  p-l), 

12.  Fundamental  Systems  of  Modular  Invariants. — ^^^lile,  by 
(22),  the  characteristic  invariants  Iq,  •  •  - ,  In-i  form  a  fundamental 
svstem  of  modular  invariants  of  a  svstem  S  of  modular  forms,  it 
is  usuallv  much  easier  to  find  another  fundamental  svstem.  In 
fact,  certain  invariants  are  usually  known  in  advance,  e.  g.,  the 
invariants  of  the  corresponding  system  of  algebraic  forms.  We 
shall  prove  the  following  fundamental  theorem: 

7/  the  modular  invariants  A,  B,  •  •  ■ ,  L  completely  characterize 
the  classes,  they  form  a  fundamental  system  of  modular  invariants. 

For  example,  7o,  •  •  • ,  In~i  evidently  completely  characterize 
the  classes  and  were  seen  to  form  a  fundamental  system. 

Let  Ci,  •  •  •,  Cs  be  the  coefficients  of  the  forms  in  the  system  <S. 
Let  each  d  take  the  values  0,  1,  •  •  •,  p  —  1.  For  the  resulting 
2?*  sets  of  values  of  the  c's,  let  the  rational  integral  functions 
A,  B,  •  •  • ,  L  oi  Ci,  •  •  • ,  Cs  take  the  distinct  sets  of  values 

Ai,    Bi,     •  •  •,     Li      (i  =  0,  •  •  •,  n  —  1). 

Thus  there  are  n  classes  of  systems  S  and  by  hypothesis  the  ith 
class  is  uniquely  defined  by  the  values  Ai,  -  ■  • ,  Li  of  our  invariants. 
A  rational  integral  invariant  0(ci,  •  •  ■ ,  Cg)  takes  the  same  value 
for  all  systems  of  forms  in  the  ith  class,  so  that  this  value  may 
be  designated  by  0,.     Now  the  polynomial 


71-1 


P{A,  5,  • .  •,  Z)  -  Z0r[l  -  (^1  -  Ai)^'}  • . .  {1  -  (Z  -  Li)^'} 

t  =  0 

is  congruent  to  0:  when  A  =  Ai,  -  -  • ,  L  =  Li  (mod  p).     Hence 

</>(ci,  •  •',  Cs)  ^  P(A,  B,  •■•,  L)         (mod  p) 

for  all  sets  of  integral  values  of  Ci,  •  •  •,  Cs.  In  view  of  Fermat's 
theorem,  we  may  assume  that  each  exponent  in  <^(ci,  •••,  Cj) 
is  less  than  jj-     If  we  replace  A,  •  •  ■ ,  L  hy  their  expressions  in 


INVAEIANTS    AND   NUMBER  THEORY.  15 

terms  of  the  c's,  P  (A,  •  •  • ,  L)  becomes  a  polynomial,  which,  after 
exponents  are  reduced  below  p,  will  be  designated  by  \p  (ci,  •  •  • ,  Cg) . 
Then  0  and  \J/  are  identically  congruent  in  Ci,  •  •  • ,  Cs,  that  is, 
corresponding  coefficients  are  congruent  modulo  i?.  In  fact,  a 
polynomial  of  type  0  is  uniquely  determined  by  its  values  for 
the  p*  sets  of  values  of  Ci,  •  •  •,  c^,  each  chosen  from  0,  1,  •  •  •, 
p  —  1  (§9).  Hence  0  can  be  expressed  as  a  polynomial  in  .1, 
•  •  • ,  L  with  integral  coefficients.* 

13.  Minor  Role  of  Modular  Covariants. — In  contrast  with  the 
case  of  algebraic  forms,  the  classes  of  modular  forms  are  com- 
pletely characterized  b}'  rational  integral  invariants.  Such 
invariants  therefore  suffice  to  express  all  invariant! ve  properties 
of  a  system  of  modular  forms.  In  this  respect,  modular  co- 
variants  play  a  superfluous  role.  For  instance,  a  projective 
property  of  a  system  of  algebraic  forms  is  often  expressed  by 
the  identical  vanishing  of  a  covariant.  But  if  C  is  a  modular 
covariant  with  the  coefficients  ci,  •  •  •,  Cs,  then  7o  given  by  (21) 
is  a  modular  invariant  of  C  and  hence  of  the  initial  system  of 
forms.  We  have  C  =  0  or  C  ^  0  (mod  p)  identically,  according 
as  7o  =  1  or  7o  =  0. 

14.  References  to  Further  Developments. — This  general  theory  of 
modular  invariants  has  been  applied  by  me  to  determine  a  com- 
plete set  of  linearly  independent  modular  invariants  of  q  linear 
forms  on  m  variables,!  and  a  fundamental  system  of  modular 
invariants  of  a  pair  of  binary  quadratic  forms  and  of  a  pair  of 
binary  forms,  one  quadratic  and  the  other  linear.| 

The  theory  has  been  extended  to  combinants  and  applied  to  a 
pair  of  binary  quadratic  forms. § 

*  This  correct  theorem  for  any  finite  field  cannot  be  extended  at  once  to 
any  field  as  stated  by  me  in  Avierican  Journal  of  Mathematics,  vol.  31  (1909), 
top  of  p.  338. 

t  Proceedings  of  the  London  Mathematical  Society,  ser.  2,  vol.  7  (1909), 
pp.  430-444. 

t  American  Journal  of  Mathematics,  vol.  31  (1909),  pp.  343-354;  cf.  pp. 
103-146,  where  a  less  effective  method  is  used. 

§  Dickson,  Quarterly  Journal  of  Mathematics,  vol.  40  (1909),  pp.  349-366. 


LECTURE  II 

SEMIXVARIANTS  OF  ALGEBRAIC  AND  MODULAR  BINARY 

FORMS 

Introductory  Exl\mple  of  the  Binary  Quartic  Form,  §§  1-6 
1.  Comparative  View. — Let  the  forms 

/  =  ao.T^  +  4ai.T^?/  +  Qaix-y-  +  ■^azxy^  +  a^y^, 

with  real  or  complex  coefficients,  be  separated  into  classes  such 
that  two  forms  /  are  transformable  into  one  another  by  a  trans- 
formation of  type 
(1)  X  =  x'  +  ty',     y  =  y', 

if  and  only  if  they  belong  to  the  same  class.  Then  a  single- 
valued  function  »S(ao,  '•,  04)  is  called  a  semin  variant  of/  if 
it  has  the  same  value  for  all  of  the  forms  in  any  class. 

By  the  repeated  application  of  this  definition  and  without  the 
aid  of  new  principles,  we  shall  obtain  a  fundamental  system  of 
rational  integral  seminvariants  of  /,  then  on  the  one  hand  the 
additional  single-valued  seminvariant  needed  to  form  with  these 
a  fundamental  system  of  single-valued  seminvarints,  and  on  the 
other  hand  the  additional  rational  integral  modular  seminvariants 
needed  to  form  with  them  a  fundamental  system  of  modular 
seminvariants  of  /.  It  is  such  a  comparative  view  that  we  desire 
to  emphasize  here.  In  later  sections,  we  shall  show  that  it  is 
usually  much  simpler  to  treat  the  modular  case  independently 
and  in  particular  without  introducing  all  of  the  algebraic  semin- 
variants, which  become  very  numerous  and  most  unwieldy  for 
forms  of  high  degree.  The  rational  integral  seminvariants 
S  of  an  algebraic  form  are  of  special  importance  since  each  is 
the  leading  coefficient  of  one  and  but  one  covariant,  which  can 
be  found  from  S  by  a  process  of  differentiation.     For  example, 

the  seminvariant  ao  is  the  leading  coefficient  of  the  covariant  /. 

16 


INVAEIANTS   AND   NUMBER  THEORY.  17 

2.  The  Classes  of  Algebraic  Quartic  Forms. — Consider  a  quartic 
form  /  in  which  ak  is  the  jBrst  non-vanishing  coefficient.  Apply 
transformation  (1)  with 

We  obtain  a  form  having  zero  in  place  of  the  former  ak+i.  Drop- 
ping the  accents  on  x',  y',  we  obtain,  for  ^'  =  0,  1,  2,  3,  the  re- 
spective forms 

(3)  ao  +  0:  a^x^  +  Qa^-^S^xY  +  "^a^-^SzX^  +  ar^S,y\ 

(4)  ao  =  0,     ai  4=  0:  Aaia?y  +  aC^Suxf  +  ar^Sui/, 

(5)  ao  =  Oi  =  0,     02  =t=  0 :  •      Qa^x^y"^  +  ^ao-^S^iy^, 

(6)  ao  =  ai  =  02  =  0,     as  =1=  0:  4a3a:?/^, 

(7)  ao  =  ai  =  a2  =  as  =  0:  aiy^, 
no  transformation  having  been  made  in  the  last  case.     Here 

(8)  ^2  =  aoao  —  a-c,     S3  =  ao^as  —  3aoaia2  +  2ai^l 

(9)  Si  =  ao^tti  —  4ao-aia3  +  Qaoai^a2  —  3ai^ 

Si3  =  4aia3  —  3a2^     Su  =  ai^ai  —  2aia2az  +  a2^ 
024  =  00204  —  203^. 

If  we  apply  to  one  of  the  forms  (3)-(6)  a  transformation  (1) 
with  i  4=  0,  we  obtain  a  form  having  an  additional  (second) 
term.  Hence  no  two  of  the  forms  (3)-(7)  can  be  transformed 
into  each  other  by  a  transformation  (1),  so  that  each  represents 
a  class  of  forms.  For  example,  there  is  a  class  (5)  for  each  set 
of  values  of  the  parameters  02  and  *S24  («2  +  0). 

3.  Rational  Integral  Seminvariants  of  an  Algebraic  Quartic. — 
First,  oo  is  a  seminvariant  since  it  has  a  definite  value  4=  0  for 
any  form  in  any  class  (3)  and  the  value  zero  for  any  form  in 
any  class  (4)-(7).  Next,  S^,  S3,  S^  are  seminvariants,  since 
they  have  constant  values 

(11)  ^2  =  -  Oi2,     S3  =  2ai\     Si=  -  3oi^       (if  Oo  =  0) 

3 


y 


18  THE   MADISON   COLLOQUIUM. 

for  any  form  in  any  class  (4)-(7),  and  constant  values  for  any 
form  of  a  definite  class  (3),  for  which  therefore  cto  has  a  definite 
value  4=  0  and  aQ~^S2,  •  •  • ,  and  hence  each  Si,  has  a  definite 
value.  Moreover,  these  seminvariants  ao,  S2,  Ss,  Si  completely 
characterize  the  classes  (3). 

Consider  a  quartic  form  /  in  which  ao,  ai,  ao,  as,  ai  are  arbi- 
trary, except  that  Oo  =1=  0.  Any  rational  integral  seminvariant 
S(ao,  •  •  •  ,ai)  has  the  same  value  for  /  as  for  the  particular  form 
(3)  in  the  same  class  as  /.     Hence 


S  =  S  [  ao,  0,  -^ ,  ~2 '  'A  I  = 
\  ao    ao     ao  / 


(f)(ao,  S2,  Sz,  S4) 


ao' 

where  ^  is  a  rational  integral  function  of  its  arguments.  We 
therefore  seek  such  functions  <f)  as  are  divisible  by  a  power  of  ao, 
and  hence  by  (11)  in  which  the  terms  involving  only  ai  cancel. 
The  function  of  lowest  degree  is  evidently 

(12)  ^4  +  3^2'  =  aoU,     I  =  ao«4  -  4aia3  +  Sas^.  / 

The  next  lowest  degree  is  6  and  the  function  is 
^^2^4  +  eSz''  +  (3d  +  4:e)S2\ 
The  coeflBcient  of  d  is  Oo^/*S2,  that  of  e  is 

Sz'  +  4^2^  =  aoW  ^' 
(13) 

(D  =  aiai  —  ^aoaxa^az  +  \.aoai  +  4ai%3  —  3a-^a'i). 

Hence  for  d  =  1,  e  =  —  1,  the  function  is  the  product  of  ao^  and 

(14)  IS2—D  =  aoJ,     J^aoa2ai^—aoa^-\-2axa2az—ai'ai—a2^. 

We  do  not  retain  D  since  it  is  expressible  in  terms  of  the  other 
functions.     Eliminating  D  between  (13)  and  (14),  we  get 

(15)  Sz"-  +  41S23  -  aoHS2  +  aoV  =  0.      1 

Now  I  and  J  are  seminvariants.  Indeed,  if  ao  =}=  0,  they  are 
expressible  in  terms  of  the  parameters  ao,  Si  in  (3)  and  hence 
each  has  the  same  value  for  any  form  in  a  class  (3) ;  while 

(16)  1=  -  Sn,     J  =  -  Su  (if  ao  =  0), 


INVAEIANTS   AND   NUMBER  THEORY.  19 

SO  that  each  has  the  same  value  for  any  form  in  a  class  (4); 
finally, 

(17)  /  =  302-,     J  =  -  ai^         (if  oo  =  ai  =  0), 

so  that  each  has  the  same  value  for  any  form  in  a  class  (5)-(7). 
From  0  we  eliminate  *S4  by  means  of  (12)  and  then  the  second 
and  higher  powers  of  Sz  by  means  of   (15).     Thus  S  equals 
Nftto'',  where  iV  is  a  rational  integral  function  of 

(18)  ao,     S2,     Sz,     I,     J, 

of  degree  0  or  1  in  Sz-  If  k  >  0,  we  may  evidently  assume  that 
not  every  term  of  the  polynomial  ^V  in  the  arguments  (18)  has 
the  factor  gq.  Let  P{S2,  Sz,  I,  J)  denote  the  aggregate  of  the 
terms  of  N  not  involving  ao  explicitly.  We  shall  prove  that, 
if  k  >  0,  N/ao''  is  then  not  a  rational  integral  function  of  ao,  •  •  • , 
tti.  For,  if  it  be,  P  vanishes  when  ao  =  0.  By  (11)  and  (16), 
the  terms  independent  of  ao  in  J  involve  a^,  while  those  in  I, 
(S2,  Sz  do  not.  Hence  J  does  not  occur  in  P.  Then,  by  (11) 
and  the  term  3a2^  in  I,  we  conclude  that  I  does  not  occur  in  P. 
Thus  P  is  a  polynomial  in  S2  and  Sz  of  degree  0  or  1  in  Sz  and 
is  not  identically  zero.     By  (11),  it  cannot  vanish  for  ao  =  0. 

Under  the  initial  assumption  that  ao  +  0,  we  have  now  proved 
that  any  rational  integral  seminvariant  S  equals  a  polynomial 
in  the  functions  (18).  The  resulting  equality  is  therefore  an 
identity. 

The  seminvariants  (18)  form  a  fundamental  system  of  rational 
integral  seminvariants  of  the  algebraic  quartic  form* 

They  are  connected  by  the  relation,  or  syzygy,  (15). 

4.  Invariantive  Characterization  of  the  Classes. — By  §  3,  the 
classes  (3)  are  completely  characterized  by  the  seminvariants 
ao,  S2,  Sz,  L  These  with  J  characterize  the  classes  (4)  having 
ao  =  0,  oi  4=  0.  For,  by  (11),  *S2  and  ^3  determine  ai;  while, 
by  (16),  /  and  J  determine  the  remaining  parameters  in  (4). 

*  The  above  proof  differs  from  that  by  Cayley  in  minor  details  and  in  the 
method  of  obtaining  the  functions  (18)  and  the  verification  that  they  are 
seminvariants  (the  present  method  being  based  upon  the  classes). 


20  THE   MADISON   COLLOQUIUM. 

The  parameter  aa  (02  4=  0)  in  (5)  is  determined  by  I  and  J,  in 
xdew  of  (17). 

We  have  now  gone  as  far  as  is  possible  in  the  characterization 
of  the  classes  by  means  of  rational  integral  seminvariants  S, 
since  the  parameters  <S24,  «3,  cla.  in  (o)-(7)  cannot  be  determined 
by  such  seminvariants.  Indeed,*  for  ao  =  «i  =  0,  we  have 
jS2  =  'S's  =  0  by  (11),  while  /  and  J  reduce  to  powers  of  a^  by  (17). 

5.  Single-valued  Seminvariants. — We  may,  however,  construct 
a  single-valued  seminvariant  which  shall  determine  these  out- 
standing parameters  *S24,  as,  a^.  To  this  end  consider  the  single- 
valued  function  V  defined  as  follows  by  its  values  in  the  sense 
of  Dirichlet.  We  take  F  =  0  if  ctq  +  0  or  if  ai  4=  0,  and  V  =  ^24, 
as,  ai  in  the  respective  cases  (5),  (6),  (7).  Since  V  has  the  same 
value  for  all  forms  in  any  class,  it  is  a  seminvariant.  The 
seminvariants  (18)  and  T'  completely  characterize  the  classes 
(3)-(7)  and  hence,  by  §  2  of  Lecture  I,  form  a  fundamental  system 
of  single-valued  seminvariants  of  the  algebraic  binarj'  quartic 
form. 

6.  Seminvariants  of  a  Modular  Quartic  Form. — Passing  to  the 
number  theory  case,  let  the  coefficients  of  the  quartic  form  / 
be  integers  taken  modulo  p,  where  p  is  a  prime  exceeding  3. 
The  denominator  in  (2)  is  then  not  divisible  by  p,  so  that  the 
classes  are  again  (3)-(7). 

By  the  general  theory  in  Lecture  I,  it  is  possible  to  character- 
ize all  of  the  classes  by  means  of  rational  integral  seminvariants, 
and  the  latter  will  then  form  a  fundamental  system.  In  par- 
ticular, we  do  not  now  require  the  use  of  such  a  bizarre  function 
as  that  used  in  §  5. 


*  A  proof  of  this  fact,  not  based  upon  the  final  theorem  of  §  3,  would  afford 
a  better  insight  into  the  nature  of  the  last  steps  in  §  3  and  explain,  in  particular, 
why  we  stopped  with  /  and  J  and  did  not  consider  combinations  of  the  Si  of 
higher  than  the  sLxth  degree  in  the  a's.  To  this  end,  let  »S  be  a  seminvariant 
homogeneous  of  total  degree  i,  in  the  a's,  and  isobaric,  of  constant  weight  w. 
As  well  known,  4i  =  2w.  Thus  S  cannot  have  a  term  03'  or  04'  and  cannot 
reduce,  when  Oo  =  Oi  =  0,  to  a2'*S24"'  {m  >  0),  of  degree  I  +  2m  and  weight 
21  +  Qm. 


INVARIANTS   AND   NUMBER   THEORY.  21 

We  shall  make  frequent  use  of  the  abbreviation 

(19)  P.  =  (1  -  ao^^)(l  -  ai^i)  • . .  (1  -  ar'). 

Then  PiS2i,  Pidz  and  Pza^  are  seminvariants*  since  each  takes 
the  same  value  for  all  forms  in  any  class.  For  the  classes  (5), 
(6)>  (7),  their  values  are  <S24,  03  and  a^,  respectively.  Hence 
the  fife  seminvariants  (18)  together  with  PiS^i,  P^dz  and  Pza^ 
completely  characterize  the  classes  and  therefore  form  a  fundamental 
system  of  rational  integral  seminvariants  of  the  quartic  form  f 
with  integral  coefficients  taken  modulo  p,  p  >  3. 

Seminvariants'  of  a  Modular  Binary  [Form   of  Order  n, 

§§  7-13 

7.  Fundamental  System  of  Modular  Seminvariants  Derived 
by  Induction  from  n  —  1  to  n. — It  is  necessary  to  distinguish  the 
case  in  which  the  modulus  p  is  prime  to  n  from  the  case  in  which 
p  divides  n.  Binomial  coefficients  for  the  form  are  not  per- 
missible in  the  second  case  and  often  not  in  the  first  case  (for 
example,  if  n  =  4,  p  =  3,  since  (2)  is  then  divisible  by  p). 
Denote  the  form  by 

(20)  Fn  =  A,x-  +  Aix^-'y  +  •  •  •  +  Any\ 

First,  let  p  be  prime  to  n.  For  Aq  4=  0,  we  can  transform  Fn 
into  a  form  lacking  the  second  term  and  having  as  coefficients 
the  quotients  of 

0-2  =  nAQA2  —  1(71  —  \)A-c, 

^^^^  <jz=n^A^Az-{n-2)nA^A^At-\-\{ii-\){n-2)A^,    ••• 

by  powers  of  nA^.  These  may  also  be  obtained  from  (8)  by 
identifying  F„  with 

n(n  —  1) 
(22)        /„  =  aox^  +  naix^-'y  +  -^ a^x^'Y  +  • ' '  • 


*  The  first  is  one-half  the  discriminant  of  the  semicovariant 
P1//2/-  =  Pi(6a2a;2  +  403x1/  +  04^")         (mod  p), 
and  the  last  two  are  the  seminvariants  of  P-Jly^  =  PiiiOiX  +  Oty)  (mod  p). 


22  THE   MADISON   COLLOQUIUM. 

For  p  prime  to  n,  a  fundamental  system  of  seminvariants  of  Fn 
is  given  by  Aq,  at,  •  •  • ,  <Tn  together  with  a  fundamental  system  of 
the  particular  form  of  order  n  —  1 

F:_,  =  PoFJy 
(23) 

=  PoAix--^i-PoA2X^-hj-\- . .  •  +PoAny''-'        {mod  p), 

where  Po=  \  —  A(r~^. 

Indeed,  Aq,  (t^,  •  -  • ,  an  completely  characterize  the  classes  of 
forms  Fn  with  ^o  =f=  0.  Since  yFn-i  =  Fn  identically,  when 
Ao  =  0,  the  classes  of  forms  Fn  with  ^o  =  0  are  completely 
characterized  by  the  seminvariants  of  the  fundamental  system 
for  Fn-i'. 

For  example,  Ao  and  PoAi  form  a  fundamental  system  of 
modular  seminvariants  of  AqX  -{-  Aiy  (since  these  characterize 
the  classes  represented  by  AqX  and  Aiy).  The  corresponding 
functions  for 

Fi'  =  PoAix  +  PoA^y   " 
are  PqAi  and 

{1  -  (PoA{)^']PoA2  =  a-  Ai^-')PoA2  =  P1A2        (mod  p). 

Hence  the  theorem  shows  that,  if  p  >  2, 

(24)  Ao,    20-2  =  4:AoA2  -  Ai\    PoAi,    P^A^ 

form  a  fundamental  system  of  modular  seminvariants  of  F2.     For 
/2,  these  are 

(24')  2ao,     S2  =  00*2  —  ai^,     Pqcii,     Pia2. 

8.  Order  a  Multiple  of  the  Modulus. — Next,  let  n  =  pq.  By 
Fermat's  theorem,  x^  —  xy^~^  and  hence 

(25)  (j)  =  Ao(xP  -  xyp-^)^ 

is  unaltered  modulo  p  by  any  transformation  (1).     Hence  if,  for 
each  value  of  the  semin variant  Aq,  we  separate  the  forms 

(26)  Fn-i  ^l(Fn-  4>) 


INVARIANTS   AND   NUMBER  THEORY.  23 

into  classes  under  (1),  multiply  each  form  by  y  and  add  0,  we 
obtain  the  classes  of  forms  F„  for  this  value  of  Aq.     Hence,  if  n 
is  divisible  by  p  a  fundamental  system  of  modular  seminvariants 
of  Fn  is  given  by  Ao  and  a  fundamental  system  for  Fn~i. 
For  example,  if  n  =  y  =  2, 

Fi  =  {A,  +  A,)x  +  Aoy     ^' 

can  be  transformed  into  x  or  Aiy  by  (1),  according  as  ^o  +  ^i  —  1 
or  0  (mod  2).  Adding  0  =  A^ix^  —  xy)  to  xy  and  A^y"^,  we  obtain 
representatives  of  the  classes  of  forms  F2.  Hence  the  6  classes 
are  completely  characterized  by  the  seminvariants  Ao  and  those 
(§  7)  of  Fi,  and  hence  by 

(27)  ^0,    ^1,    J  =  (1  +  ^0  +  Ai)Ai. 

9.  Seminvariants  of  the  Binary  Cubic  Form. — The  classes  of 
algebraic  forms  fz  are 

(28)  a,x^  +  ^a^-'Sixy''  +  a^-^Szf, 

(29)  Zaix^y  +  \ar^Sizf,     Za^xy"",     a^f, 

where  the  *S's  are  given  by  (8)  and  (lOi).  The  discriminant  D 
of/3  is  given  by  (13).  As  in  §  3,  ao,  'S2,  Sz,  D  form  a  fundamental 
system  of  seminvariants  of /a;  they  are  connected  by  the  syzygy 

(13). 

Henceforth,  let  the  coefficients  of /a  be  integers  taken  modulo  p, 
the  excluded  case  p  =  3  being  treated  in  §  15.  If  p  >  3,  the 
classes  are  again  (28)  and  (29),  and  a  fundamental  system  of 
seminvariants  is  given  by 

(30)  ao,     S2,     S3,     D,     Piao,     P^az. 

It  is  instructive  to  compare  this  result  with  that  obtained  by 
the  method  of  §  7.     Forming  the  functions  (24)  for 

/2'  =  Po/3/2/  =  3Poaia:2  +  ?>Poa2xy  +  Poas/        (mod  y), 

and  deleting  the  factor  3  from  the  first  and  second,  we  get* 

Poai,     5  =  Po(4aia3  —  ^at^)  =  P^Su,     Pia%,     P^az. 


*  They  characterize  the  classes  (29)  of /s  with  Co  =  0  and  may  be  so  derived. 


24  THE   MADISON   COLLOQUIUM. 

Hence,  if  p  >  3,  these  four  functions  and  uq,  S2,  S3  form  a  funda- 
mental system  of  modular  seminvariants  of  /s.  We  may  drop 
Potti  since 

(31)  PqS2  '   Ss=±  2Poai^  =  ±  2Poai        (mod  p). 
Hence  a  fundamental  system  of  seminvariants  of  /s  for  2?  >  3  is 

(32)  ao,     S2,     Ss,     d  =  Po5i3,     Pia2,     P2az. 

It  is  easy  to  deduce  5  from  the  old  set  (30),  and  D  from  this  new 
set.* 

Finally,  let  p  =  2.  By  §  7,  a  fundamental  system  of  sem- 
invariants for  fz  is  given  by  a^,  S2,  S3  and  a  fundamental  system 
for  /2'.  The  latter  system  is  derived  from  (27)  by  replacing 
Ao,  Ai,  A2  by  Potti,  Poa2,  P0O3,  and  hence  is 

(1  +  ao)ai,     (1  +  ao)a2,     (1  +  flo)(l  +  ai  +  a2)a3. 

We  may  drop  (1  +  00)^1  —  (1  +  ao)S2. 

10.   The  Binary  Quartic  Form.     For  p  =  2,  we  have 

F3  =  Aio^  +  (^0  +  A2)x'y  +  A3X2/  +  A,f, 

whose  seminvariants  are  obtained  from  those  of  /a  at  the  end  of 
§  9.     They  with  Ao  give  a  fundamental  system  of  seminvariants 

of  P4: 

Ao,    Ai,    A1A3  +  Ao  +  A2,     (1  +  Ai)A3, 

AiA,  +  AiAziAo  +  A2),    K  =  (1  +  ^0(1  +  ^0  +  ^2  +  A3)A,, 

An  equivalent  fundamental  system  isf 

Ao,    Ai,    A2  +  A3,     (l  +  Ai)A2, 
(33) 

^1^44  +  A0A2  +  A2A3,    K. 


*  D  =  aoP-'iSi"  +  45/)  -  dS2        (mod  p). 

For,  if  Oo  4=  0,  then  5^0  and  this  relation  follows  from  (13);  while,  if  ao  =  0, 
D  =  Ui^Siz  =  ai^S  =  —  Sib.  Conversely,  5  can  be  expressed  in  terms  of  the 
functions  (30).  The  above  relation  gives  5^2.  The  product  of  this  by  S^p'^ 
is  congruent  to  5  if  (S2  +  0.  Also  5  =  0  if  co  +  0.  There  remains  the  case 
in  which  Si  =  0,  Oo  =  0,  whence  ai  =  0,  5  =  —  Saa^  =  —  3(Fia2)^. 

t  Annals  of  Malhemalics,  ser.  2,  vol.  15,  March,  1914.     I  there  give  also  a 
complete  set  of  linearly  independent  invariants  and  of  linear  covariants 


INVARIANTS   AND   NUMBER  THEORY.  25 

For  p  >  3,  fz  is  obtained  from  fz  by  replacing  Qq,  ai,  a^y  az  by 
4aiPo,     2a2Po,     3«3-Po,     ^4^0, 

respectively.  Making  this  replacement  in  the  second  set  of  sem- 
invariants  of  fz  in  §  9,  we  obtain  P^ai,  which  may  be  dropped  in 
view  of  (31),  and  the  last  five  functions  (34).  Hence, /or  p  >  3, 
a  fundamental  system  of  modular  seminvariants  of  f^  is  given  by 

(34)  floj    S2,    Sz,    Si,   PqSiz,   PoSii,    P1024,    P2O3,    Pzdi' 

Here  the  three  Sij  are  given  by  (10).  Since  the  functions  (34) 
completely  characterize  the  classes  (3)-(7),  we  have  a  new  proof 
that  they  form  a  fundamental  system. 

11.  Explicit  Fundamental  System  ivhen  p  >  n. — Instead  of 
employing  the  above  step  by  step  process,  we  can  obtain  directly 
a  fundamental  system  of  modular  seminvariants  of  fn  when  the 
modulus  p  exceeds  the  order  n  of  the  binary  form  (22).  Consider 
a  particular /n  in  which  ak  is  the  first  non-vanishing  coefficient: 

Z  (  ^  )  ttiX'^-y  (flk  +  0). 

i=k  \  *  / 

To  this  we  apply  transformation  (1)  and  obtain 

i=k    j=0    \  ^    /     \  3  J  l=k 

where  we  have  replaced  j  by  I  —  i  and  set 

-"=s(:)C;-";)-''--(:)s(9-'-- 

Take  k  <  n  and  give  to  t  the  value  (2).     Thus 

(n\ 

(35)  "^'^  "  {{k+l)auV-' ' 


26  THE   MADISON   COLLOQUIUM. 

In  particular, 


(^kk  —    Ij       <^kk+\  =  0,       <Tqi 


=  T.^{-iy-'{^^a^-w-%, 


the  last  being  the  algebraic  seminvariant  designated  earlier  by 
Si.  It  is  obtained  from  the  expansion  of  (ao  —  clxY  by  replacing 
a  single  ao  in  each  term  by  ai.  Except  for  a  numerical  factor 
not  divisible  by  -p,  <Jki  (for  0  <  A;  <  Z  —  1)  equals  the  Ski  in  (10) 
and  in  (38)  below. 

The  classes  C^  of  forms /„  in  which  au  is  the  first  non-vanishing 
a  are  distinguished  from  each  other  by  the  value  of  a„  if  A:  =  n, 
and  if  ^  <  n  by  the  values  of  the  parameters  at,  dki  {I  =  k  -{-  2, 
•■',  n).  Employing  the  notation  (19),  we  shall  verify  that 
Pk-icik  and  Pk-\<Jki  are  modular  semin variants  of  /„.  They 
vanish  for  a  form  Cj  (j  ^  ^  —  1)  since  then  1  —  af~^  =  0. 
For  Ck,  they  reduce  to  the  parameters  ak  and  aki  of  that  class. 
For  flo  =  0,  •  •  • ,  ttk  =  0,  the  first  is  zero  and  the  second  is  the 
expression  for  aki  when  Uk  =  0,  whose  non-vanishing  terms 
(given  by  i  =  k  and  i  =  k  -\-  1)  are  constant  multiples  of 
al.+t;  but  ak+i  is  constant  for  any  class  Cj  (j  >  k). 

It  follows  also  that  the  parameter  ak+\  in  a  class  Ck+i  is  de- 
termined by  the  semin  variants  Pk-iO'ki  (I  =  k  -{-  2,  k  -{-  3), 
provided  k  -{-  3  ^  n.  But  a„_i  and  a„,  not  so  determined,  are 
found  from  Pk-idk  (k  =  n  —  1,  n).  Hence  a  fundamental 
system  of  modular  seminvariants  of  fn,  for  p  >  n,  is  given  by 

ao,     (Xoi  (/  =  2,  •  •  •,  n), 

(36)  Pk-KTki     {k=l,---,n-2',l  =  k-\-2,  ■'•,n), 

Pn—20'n—l,       Pn—lttri' 

For  71  =  2,  3,  4,  these  are  (24'),  (32),  (34),  respectively,  except 
for  the  difference  of  notation  indicated  above.  For  n  =  5,  we 
see  that  a  fundamental  system  of  modular  seminvariants  of  /s, 

for  p  >  5,  is 

■\ 

ao,     S-z,     S3,     04,     05,     PqSis,     PoSu,     P0S15, 
(37) 

PlS2i,      PlS2i,       P2S35,       Psdi,       P^db, 


INVARIANTS  AND   NUMBER  THEORY.  27 

in  ivhich  the  symbols  are  defined  by  (8)-(10),  (19)  and 

/So  [=  ao%5  —  5ao^aiai  +  10ao^«i^«3  —  10aoai^a2  +  4a i^ 

*Si5  =  16a  1^05  —  40ai%2a4  +  40aia2^a3  —  15a2^ 
(38) 

Sih  =  27a2-a5  —  45a2a3a4  +  20a3^ 

Sz5  =  SasOs  —  5a4^ 

12.  Another  Method  for  the  Case  p  >  ??. — We  may  formulate  the 
method  of  §  7  so  that  it  shall  be  free  from  the  induction  process. 
The  classes  of  forms  (23)  with  PoAi  4=  0,  and  hence  the  classes 
of  forms  Fn  with  .4o  =  0,  Ai  ^  0,  are  characterized  by  the 
seminvariants  given  by  the  products  of  Po  by  the  functions 
az',  •  • '  obtained  from  0-2,  0-3,  •  •  • ,  (Xn-i  by  increasing  the  subscript 
of  each  Ai  by  unity  and  replacing  nhy  n  —  1;  indeed,  Pi^  =  Pi 
(mod  p).  When  the  process  of  deriving  (23)  from  (20)  is  applied 
to  (23),  we  get 

F'U  =  [1  -  (PoA,)^']F:_Jy  =  (1  -  A,^-')PoFjf 

(39)  =  PiFnhf  ^  PiA^x^-'  +  PiAzx--'y 

H hPi^n?/"-'         (mod  2?). 

The  class  of  forms  (39)  with  P1A2  +  0,  and  hence  the  classes  of 
forms  Fn  with  Ao  =  Ai  =  0,  A2  4=  0,  are  characterized  by  the 
seminvariants  given  by  the  products  of  Pi  by  the  functions  0-2", 
•  •  •  obtained  from  a^,  •  •  • ,  (rn-2'  by  increasing  the  subscript  of 
each  Ai  by  unity  and  replacing  nhy  n—  1.  Finally,  we  obtain 
P„_2.4„_i.r  +  Pn-2Any,  characterized  by  the  seminvariants 
Pn-2^n-i  and  Pn~iAn.     The  earlier  Pk-iAk   may  be   dropped 

(§  11). 
For  example,  if  n  =  3,  p  >  3,  the  fundamental  system  of  F3  is 

Ao,     a2,     as,     Po<T2    =  Po(4^i^3  -  ^2'),     Pl^2,     P2^3. 

Changing  the  notation  from  Fz  to  fs,  we  see  that  0-2'  becomes 
3(4aia3  —  3a2-),  so  that  the  resulting  seminvariants  are  (32). 
We  may  of  course  apply  the  method  directly  to/3;  in  S2  we  replace 
oo,  ai,  a2  by  3ai,  fa^,  03  and  obtain  |(4aia3  —  302"). 


28  THE    MADISON    COLLOQrir:M. 

Again,  to  find  a  fundamental  system  of  /4  for  p  >  o,  we  take 
ao,  S2,  S3,  Si  and  the  products  of  Po  by  the  functions  f <Si3  and 
16*514  obtained  from  S2  and  S3  by  replacing  ao,  cii,  ^2,  0,3  by 
4a  1,  ^  •  6a2,  -j  •  4a3,  a4;  then  the  product  of  Pi  by  the  function 
2S24:  obtained  from  S2  by  replacing  ao,  ai,  ao  by  6a2,  ^  •  4a3,  Ui; 
then  Pofls  and  P3a4,  to  characterize  P2(4a3a:  +  04?/).  We  again 
have  (34). 

13.  Number  of  Linearly  Independent  Seminvariants. — Let 
p>  n  and  employ  the  notations  of  §  1 1 .  In  the  classes  Ck  (k<n). 
Akk  =  ak  (;;)  has  p  —  1  values,  ^^-jt+i  =  0,  while  Akk+2,  •■,  Akn 
take  independently  the  values  0,  1,  •  •  •,  p  —  1.  In  the  classes 
Cn,  an  has  p  values.     Hence  there  are 

M-l 

P+Jl(p—  1)2?""^'"^  =  2?  +  P"  -  1 

distinct  classes  of  forms /„.  Thus  by  §  11  of  Lecture  I,  there  are 
exactly  p"^  -\-  p  —  1  linearly  independent  modular  seminvariants  of 
fn  when  p  >  n. 

Deeivatiox*  of  ^Modular  Invariants  from  Seminvariants, 

§§  14-15 

14.  Invariants  of  the  Binary  Quadratic  Form. — First,  let  p=2. 
Any  pol\'nomial  in  the  seminvariants  (27)  is  a  linear  function  of 

1,     Aq,     Ai,     AqAi,     J,     AqJ  =  A0A1A2, 

since  (^0  +  Ai)J  =  0.  Since  there  were  six  classes,  these  six 
seminvariants  form  a  complete  set  of  linearly  independent  sem- 
invariants. Now  a  seminvariant  is  an  invariant  if  and  only  if 
it  is  symmetrical  in  ylo  and  .42.     But 

I={l-Ao)a-Ai)(l-A2)^(l-Ao)(J+l+A,)         (mod  2). 

Thus  1,  Ai,  AqJ  and  /  are  invariants.     By  subtracting  constant 

*  WMle  this  method  is  usually  longer  than  the  ntiethod  of  Lecture  I,  it 
requires  no  artifices  and  makes  no  use  of  the  technical  theory  of  numbers. 
Moreover,  it  leads  to  the  actual  expressions  of  the  invariants  in  terms  of  the 
seminvariants  of  a  fundamental  system,  thus  j-ielding  material  of  value  in  the 
construction  of  covariants. 


INVARIANTS   AND   NUMBER  THEORY.  29 

multiples  of  these  four,  any  seminvariant  can  be  reduced  to 
cAo  +  dAoAi,  which  is  an  invariant  only  when  identically  zero. 
Hence  1,  Ai,  A0A1A2  and  I  form  a  complete  set  of  linearly  inde- 
pendent invariants  of  F2  modulo  2. 

Next,  let  p  >  2.  The  discriminant  of  f2  is  D  =  82-  Any 
polynomial  in  the  four  fundamental  seminvariants  (24')  is  a 
linear  function  of 

ao'D^',     Poai\     Pia2'     {i,  i  =  0,  1,  •  •  •,  p  -  1), 

since  the  product  of  PoQi  or  PiOa  by  ao  is  zero,  that  of  Pia2  by 
PoQi  or  D  is  zero,  while  DP oai  =  —  Pai^.     Further, 

Po  =  1  -  ao^\     PoW^'  -  (-  am  =  0, 

Pi  =  Po  -  Poai^\     ao^'D^  ^  D^'  -  (-  l)^Poai'', 

modulo  p.     Hence  any  seminvariant  is  a  linear  function  of 

ao^\     aoW^     (i=0,l,  •■-,p-2;j  =  0,l,  '■',p-l), 

^^^^  Poa,\     Pia2^  {k=l,  '",p-l). 

The  number  of  these  is  p^  +  p  —  1.  Hence  (§  13)  they  form  a 
complete  set  of  linearly  independent  modidar  seminvariants  of  fi 
for  p>  2. 

The  invariant  ^  =  yli  in  §  6  of  Lecture  I  becomes  for  two 
variables 

(41)      ^={ao"  +  a2"(l-ao^^)}(l-Z>^^)  =  ao"(l-^^')+^i«2% 

where  /z  =  (p  —  l)/2.     By  the  expansion  of  Dp~^,  we  get* 


(42) 


A  =  (flo'^  +  02'')  ( 1  -  I^  a^Wai^'^-'^'J  . 


*  Transactions  of  the  American  Malhemalical  Society,  vol.  10  (1909),  p.  132. 
To  give  a  direct  proof  of  the  identity  of  the  final  expression  (41)  and  (42), 
note  that  the  product  of  the  final  factor  in  (42)  by  D  equals  OoOz  —  {aoa2)i^'^^ 
algebraically,  so  that  the  product  AD  is  divisible  by  p.  But  the  product  of 
(41)  by  Z)  is  evidently  divisible  by  p.  It  therefore  remains  only  to  treat  the 
case  D  =0.  Replacmg  ai^  by  aoa2,  we  see  that  the  final  factor  in  (42)  becomes 
I  —  (jj.  +  l)ao'^C2*'.    Hence  (41)  and  (42)  are  now  identical  if 

ao'^aa" (00**  —  02*^)  =  0  (mod  p). 

But,  if  aoUi  =t=  0,  Oo'^Cz'^  =  Ci^*^  =  1,  ao*^  =  azi^  =  ±1. 


30  THE   MADISON   COLLOQUIUM. 

Since  (42)  is  therefore  a  seminvariant  and  is  symmetrical  in 
ffo  and  a2  and  since  the  weight  of  every  term  is  divisible  by 
p  —  1,  ^  is  an  absolute  invariant.     By  (41), 

A"  =  flo"^  (1  -  D^^)  +  Pla22^     (1  -  flo^')Z)^i  =  Poai^\ 
(43) 

A^  +  D^'-l^-Io,     /o=(l-«o^')(l-ai^')(l-a2^'). 

Hence  also  Iq  is  an  absolute  invariant.     Subtracting  multiples  of 

I,=  l-ao^'-Poai^'-Piao^\      A,     D^     {j=0,  1,  •  •  •,  p-1), 

we  may  reduce  any  seminvariant  to  a  linear  function  of  the  ex- 
pressions (40)  other  than  Pirt2^',  Pia2'',  D^'  (j  =  0,  -  •  -,  p  —  1). 
The  resulting  linear  function  L  is  not  an  invariant.  For  example, 
if  2?  =  3,  it  is 

L=aao'^-]-bao-{-caoD-\-daoD~-{-ePoCii-{-fPo(ti'    {a,  '  •  •,/ constants). 

Interchange  ao  and  02,  and  change  the  sign  of  Ui.     We  get 

aa2^  +  ba2  +  ca2D  +  c?a2^^  +  (1  —  02")  (/a  1^  —  eai). 

This  is  to  be  identically  congruent  to  the  invariant  L.  Taking 
02  =  0,  we  see  that  e=f=a=b  =  0,  c  =  d.  Then  L 
=  caQa2(ao  +  a2)  +  ca^a-^a2  is  not  symmetric  in  Qq  and  02. 
Hence  i  =  0.  For  any  p,  a  like  result  may  be  proved  by  con- 
sidering separately  the  terms  of  L  of  constant  weights  modulo 
p  —  \.  Hence  in  accord  with  §  11  of  Lecture  I,  a  complete  set 
of  linearly  independent  invariants  of  /2,  for  p  >  2,  is  given  by  Iq, 
A  and  the  powers  of  D.  In  place  oi  D^  =  1 ,  we  may  use  A^,  in 
view  of  (43). 

15.  Invariants  of  the  Binary  Cubic  Modulo  3. — A  fundamental 
system  of  seminvariants  of  Fz  modulo  3  is  given  by  Aq  and  a 
fundamental  system  of 

F2  =  Aix'  +  (Ao  +  A2)xy  +  Azif. 

Hence,  by  (24),  a  fundamental  system  for  F3  is  given  by 

Ao,    Ai,    t  =  AyAz  -  (Ao  +  ^2)-,     (1  -  Ai')(Ao  +  A2), 

u=  (l-^i2)[l-  (^0  +  .•l2)-M3. 


nn'AEIAXTS   AND   NUMBER  THEORY.  31 

In  place  of  the  fourth  and  third  we  may  evidently  use 

X  =    (1   -   ^li-)^o,       0-  =    ^1^13  +  A0A2  -   AM2'  =   i  +  ^0'  +  v. 

Here  a  is  the  discriminant  of  F3  for  p  =  3.  By  §  13  there  are 
11  classes  of  forms  F^.  Hence,  by  §8,  there  are  3-11  classes  of 
forms  Fz.  Thus  there  are  exactly  33  linearly  independent 
seminvariants  of  F3.     Since 

A{\  ^  Aifx  =0,    (tX  =  ^oX',    mCo-  +  ^0')  =  0, 

;,(X  +  ^0)  =  0,     (1  -  Ai^)(r  =  .4oX, 

modulo  3,  any  polynomial  in  the  seminvariants  Aq,  A\,  a,  X,  /x 
of  the  fundamental  system  is  congruent  to  a  linear  function  of 

(44)  Aq^Ai\  Aq'(j\  Ao'Ai<7\  /lo^V,  ^oV    (i,  j=0,  1,  2;  k=  1,  2). 

Hence  these  33  functions  form  a  complete  set  of  linearly  inde- 
pendent seminvariants  of  Fz.     The  seminvariants 

P  =  1  -  ^1^  _  x2  =  (1  _  A,')(l  -  A2'), 

(45)  7o  =  (1  -  Ao')(P  -  M^)  =  n  (1  -  A,'), 

t=0 

E  =  AoAi(<x  -  O  +  ^o/x  =  AoAziAoAo  -  AiAz+Ai'-A.') 

are  seen  to  be  invariants  as  follows.*  The  weights  of  the  terms  of 
each  are  all  even  or  all  odd.  Moreover,  under  the  substitution 
(^0^3) (^1^2),  induced  upon  the  coefficients  of  Fz  bj'  the 
interchange  of  x  and  y,  the  functions  a,  P  and  /o  are  unaltered, 
while  E  is  changed  in  sign.  Hence  a,  P,  Iq  are  absolute  in- 
variants, while  E  is  an  invariant  of  index  unity.  We  now  have 
7  linearly  independent  invariants 

(46)  h,     E,    E\     a,     <j\     P,     1. 
Noting  that 

(47)  E"  =  AoV  +  Ao\a-  -  a' +  V)  -  Ao\ 


*  Or  by  general  theorems,  Transactions  of  the  American  Mathematical 
Society,  vol.  8  (1907),  pp.  206-207.  Note  that  E  is  the  elimmant  of  F3  =  0, 
7?  =  X,  y^  =  y  (mod  3). 


32  THE    MADISOX   COLLOQUIUM. 

we  may  employ  the  functions  (46)  to  delete  from  (44) 

in  turn  (no  one  of  these  terms  being  reintroduced  at  a  later  stage) . 
There  remain  11  semin variants  of  odd  weight 

(48)  Ao'Au    Ao^Aicr,    Ao'Akt'-,    ju,     ^oV        {i  =  0,  1,  2), 
and  15  of  even  weight 

(49)  Ao,  Ao\  AMi\  Aoa,  Ao(t\  Ao'ct,  ^oV,  ^o'X,  Ao\',  ^o'X',  Aoix\ 

Now  the  weight  and  index  of  a  semin  variant  of  Fz  modulo  3 
are  both  even  or  both  odd.*  A  linear  combination  of  the  func- 
tions (48)  which  is  changed  in  sign  by  the  substitution  (A0A3) 
(A1A2)  is  seen  to  be  identically  zero  (it  suffices  to  set  A3  =  0, 
^2  =  0  in  turn).  A  linear  combination  of  the  functions  (49) 
which  is  unaltered  by  that  substitution  is  seen  similarly  to  be 
identically  zero.  Hencef  a  complete  set  of  linearly  independent 
invariants  of  F3  modulo  3  is  given  by  (46). 

*  When  the  sign  of  y  is  changed,  a  seminvariant  is  unaltered  or  changed  in 
sign  according  as  its  weight  is  even  or  odd. 

t  Another  proof,  using  the  classes  of  Fz  under  the  group  of  aU  binary  linear 
transformations  of  determinant  unity  modulo  3,  and  involving  a  use  of  more 
technical  theory  of  numbers,  is  given  in  Transactions  of  the  American  Mathe- 
matical  Society,  vol.  10  (1909),  pp.  149-154.  The  case  of  any  modulus  p 
is  there  treated. 


LECTURE  III 

INVARIANTS  OF  A  MODULAR  GROUP.       FORMAL  INVARIANTS 
AND   COVARIANTS  OF  MODULAR  FORMS.    APPLICATIONS 

Invariants  of  Certain  Modular  Groups,  §§  1-4 

1.  Introduction. — Let  G  be  any  given  group  of  g  linear  homo- 
geneous transformations  on  the  indeterminates  Xi,  •  •  • ,  Xm  with 
integral  coefficients  taken  modulo  y,  a  prime.  Hurwitz*  raised 
the  question  of  the  existence  of  a  finite  fundamental  sj^stem  of 
invariants  of  G.  For  the  relatively  unimportant  case  in  which  g 
is  not  divisible  by  p,  he  readily  obtained  an  affirmative  answer 
by  use  of  Hilbert's  well  known  theorem  on  a  set  of  homogeneous 
functions,  but  emphasized  the  difficulty  of  the  problem  in  the 
general  case. 

In  §  5  I  shall  consider  the  relation  of  this  question  to  that  of 
modular  covariants  and  formal  invariants  of  a  system  of  forms 
and  incidentally  answer  the  above  question  for  special  groups 
of  orders  divisible  by  jp. 

I  shall,  however,  first  present  a  simplification  of  my  own  work 
on  the  total  group.  Its  invariants  are  universal  covariants,  i.  e., 
covariants  of  any  system  of  modular  forms  (§  13).  It  was  from 
the  latter  standpoint  that  I  was  led  to  the  subject  of  invariants 
of  a  modular  group  independently  of  Hurwitz's  paper,  in  the 
title  of  which  the  word  invariant  does  not  occur. 

2.  Invariants  of  the  Total  Binary  Group. — Consider  the  group 
G  of  all  modular  linear  homogeneous  transformations  with  integral 
coefficients  of  determinant  unity: 

(1)     x'  =  bx  +  dy,     y'  =  ex  +  ey,     he  —  cd=  1         (mod  p). 

The  term  point  will  be  used  in  the  sense  of  homogeneous 
coordinates,  so  that  (x,  y)  =  {ax,  ay),  while  (0,  0)  is  excluded. 

*  Archiv  der  Mathematik  und  Physik,  (3),  vol.  5  (1903),  p.  25. 
4  33 


34  THE   MADISON   COLLOQUIU^VI. 

We  do  not  restrict  the  coordinates  to  be  integers,  but  permit 
their  ratio  to  be  a  root  of  any  congruence  with  integral  coefficients 
modulo  p.  A  point  is  called  real  if  the  ratio  of  its  coordinates  is 
rational. 

A  point  (x,  y)  is  invariant  under  a  transformation  (1)  if 
x'  =  px,  y'  =  py,  or 

(2)  (6  —  p)x  +  rf?/  =  0,     cx-\-  {e  —  p)y  =  0        (mod  p). 
If  these  congruences  hold  identically  as  to  x,  y,  then 

d=c=0,     6=e=±l         (mod  p) 
and  the  transformation  is  one  of  the  transformations 

(3)  x'  =  ±x,     y'  =  ±y         (mod  p), 

which  leave  every  point  invariant. 

A  special  point  is  one  invariant  under  at  least  one  trans- 
formation (1)  not  of  the  form  (3).  There  are  pip"^  —  1)  trans- 
formations (1).  We  shall  assume  in  the  text  that  p  >  2  (rele- 
gating to  foot-notes  the  modifications  to  be  made  when  p  =  2). 
Then  there  are  two  transformations  (3).  Hence  any  non-special 
point  is  one  of  exactly* 

(4)  CO  =  ip(p2  -  1) 

conjugate  points  under  the  group  G,  while  a  special  point  is  one 
of  fewer  than  w  conjugates. 

Let  {x,  y)  be  a  special  point  and  let  (1)  be  a  transformation, 
not  of  the  form  (3),  which  leaves  it  invariant.  Thus  the  con- 
gruences (2)  are  not  both  identities.  The  determinant  of  their 
coefficients  must  therefore  be  divisible  by  p.  Hence  p  is  a  root 
of  the  characteristic  congruence  (in  which  a  =  6  +  e) 

(5)  p^  —  ap  +  1  =  0         (mod  p). 

First,  suppose  that  (5)  has  an  integral  root  p.  For  this  value 
of  p,  one  of  the  congruences  (2)  is  a  consequence  of  the  other, 
and  the  ratio  a:  :  ?/  is  uniquely  determined  as  an  integer  modulo  p. 

*  For  p  =  2,  w  is  to  be  replaced  by  2(2^  —  1)  =6. 


INVARIANTS   AND   NmiBER   THEORY.  35 

Hence  only  real  special  points  are  invariant  under  a  transforma- 
tion [other  than  (3)]  whose  characteristic  congruence  has  an 
integral  root.  Moreover,  all  real  points  are  conjugate  under  the 
group  G.     Indeed, 

x'  =  bx,     y'  ~  x-i-  b~^y,     and     x'  =  —  y,     y'  ^  x 

replace  (1,  0)  by  {h,  1)  and  (0,  1)  respectively.     Hence  if  an 
invariant  of  G  vanishes  for  one  of  the  real  points,  it  vanishes  for 
all  and  has  the  factor 
p-i 

(6)  X  =  2/  n  (--c  -  ay)  =  x^y  -  xy^        (mod  p), 

a=0 

the  congruence  following  from  Fermat's  theorem.  Obviously, 
any  transformation  of  G  replaces  a  real  point  by  a  real  point,  and 
therefore  L  by  kL.  The  constant  k  is  in  fact  unity  and  L  is  an 
invariant  of  G.     Indeed,  for 

(7)  x^aX+hY,    y^cX+dY        (mod  p), 
where  a,  ''•,<!  are  integers  of  determinant  ^  =  ad  —  he, 

xP   yP 


(8) 


y 


aX^^hY^  cX^  +  dY^ 

aX+hY    cX+dY    =^   -     -         ^"'^^^" 


Xp  yp 

=  A 

X    Y 

Next,  suppose  that  (5)  has  no  integral  root  and  therefore  two 
Galois  imaginary  roots.  By  (2),  each  root  p  uniquely  determines 
a  point  (x,  y)  with  ?/  =1=  0.  We  may  therefore  take  ?/  =  1, 
whence  ex  =  p  —  e.  The  resulting  two  special  points  are 
therefore  imaginary  points  of  the  form  (rp  -}-  s,  1),  where  r  and  s 
are  integers  modulo  p,  and  r  is  not  divisible  by  p.  The  imagi- 
naries  introduced*  by  new  transformations  are  expressible 
hnearly  in  terms  of  this  p.  Indeed,  (2p  —  a)~  =  A,  where 
A  =  a^  —  4:  is  a  quadratic  non-residue  of  p  (i.  e.,  is  not  the  re- 
mainder when  the  square  of  any  integer  is  divided  by  p).  Thus 
A  =  a^v,  where  j'  is  a  fixed  non-residue  of  p.  Hence  the  roots 
of  all  congruences  (5)  having  no  integral  roots  are  expressible 
in  the  form  k  +  Iv^v,  where  k  and  I  are  integers. 

*  There  are  no  new  ones  if  p  =  2,  since  a  s  0  (mod  2). 


36  THE  MADISON   COLLOQUIUM. 

Hence  the  special  points  invariant  under  transformations 
whose  characteristic  congruences  have  no  integral  roots  are  all 
of  the  form  (rp  -\-  s,  1),  where  r  and  s  are  integers,  r  not  divisible 
by  2^}  while  p  is  a  fixed  root  of  a  particular  one  of  these  congru- 
ences (5). 

We  next  show  that  these  p^  —  p  imaginary  special  points  are 
all  conjugate  under  the  group  G.  It  suffices  to  prove  that  they 
are  all  conjugate  with  (p,  1),  which  is  invariant  under 

x'  =  ax  —  y,    y'  =  X. 

Now  transformation  (1)  replaces  (p,  1)  by  {R,  1),  where 

p      bp+d 
cp  +  e 

We  are  to  prove  that  there  exist  integers  h,  c,  d,  e  satisfying 

(9)  be  —  cd=  1         (mod  p), 

such  that  R^  rp  -{-  s,  where  r  and  s  are  any  assigned  integers 
for  which  r  is  not  divisible  by  p.  Denote  the  second  root  of  (5) 
by  p'  and  multiply  the  numerator  and  denominator  of  R  by 
cp'  +  e.     Using  (9),  we  get 

R  =  — ,     N  =  be  -\-  de  -J-  dca,     q  =  c~  -{-  ace  +  e^. 

We  first  show*  that  we  can  choose  integers  c  and  e  such  that 
q  ^  i  (mod  p),  where  i  is  any  assigned  integer  not  divisible  by  p. 
If  i  is  a  quadratic  residue  of  p,  we  may  take  c  =  0.  Next,  let 
i  be  a  quadratic  non-residue  of  p.  Taking  c  ^  0,  e  ^  kc,  we 
have 

q  ^  c'm,    m  =  l  +  ak  +  k\ 

Now/(A-)  =  f{K)  if  and  only  li  K^  k  ov  K^  —a  —  k.  Hence 
the  p  —  1  values  of  k  other  than  —  a/2  give  by  pairs  the  same 
value  of /(A-).  Thus  for  ^  =  0,  •  •  •,??  -  IJik)  takes  1  +  f(p-l) 
incongruent  values,  no  one  a  multiple  of  p  [since  (5)  has  no 
*  If  p  =  2,  then  a  =  Q;  taking  c  =  1,  e  =  0,  we  have  q  =  \  ^  i  (mod  2). 


INVARIANTS   AND   NUIVIBER  THEORY.  37 

integral  root],  and  consequently  a  value  which  is  a  quadratic 
non-residue  of  p.  Then,  by  choice  of  c,  q  can  be  made  congruent 
to  any  assigned  non-residue. 

Having  made  q  ^  i  (mod  p)  by  choice  of  c  and  e,  we  proceed 
to  choose  integral  solutions  h  and  d  of  (9)  such  that  N  will  be 
congruent  to  any  assigned  integer  j.  If  c  ^  0,  so  that  e  ^  0, 
we  take  d  =  j/e.  If  c  ^  0,  we  eliminate  d  from  N  by  use  of  (9) 
and  obtain 

N  ^  -{bq  —  e  —  ca),     q  =  c~  -\-  ace  +  e^. 

Since  g  ^  0,  we  may  make  N  =  j  hy  choice  of  6. 

We  have  therefore  proved  that  there  are  exactly  p"^  —  p 
imaginary  special  points,  viz.,  irp  +  s,  1),  r  ^  0,  and  that  they 
are  all  conjugate  under  the  group  G.  Hence  any  invariant  of  G 
which  vanishes  for  an  imaginary  special  point  has  the  factor 

(10)  '2  =  -V^=x.^r^. 

Indeed,  the  numerator  of  the  first  fraction  vanishes  for  x=rp-\-s, 
y=  1,  since 

{rp  +  sY'  =  rp^'  +  s,     p^'  ^  p         (mod  p), 

the  last  congruence*  being  a  case  of  Galois's  generalization  of 
Fermat's  theorem.  We  have  divided  out  L,  which  vanishes  for 
the  real  points  (s,  1)  and  (1,  0).  Since  any  transformation  of  G 
replaces  one  of  our  imaginary  points  by  another,  it  replaces  Q 
b}'-  JcQ.  The  constant  k  is  in  fact  unity  and  Q  is  an  invariant  of 
G.  Indeed,  (8)  holds  if  we  replace  the  exponents  p  by  p^. 
Hence  the  quotient  Q  is  invariantf  under  all  transformations  (7). 

*  It  may  be  proved  by  noting  that  (5)  implies 

(p2  —  ap  +  1)P  =  p-P  —  apT  +  1  3  0  (mod  Tp), 

SO  that  pP  is  the  second  root  of  (o).  By  the  same  argument,  (p^)"  is  a  root, 
distinct  from  pp,  and  hence  identical  with  p. 

1 1  gave  the  notation  Q  to  the  invariant  (10)  sincQ  it  is  the  product  of  all 
of  the  binary  quadratic  forms  x^  +  •  •  •  which  are  irreducible  modulo  -p. 
Indeed,  the  latter  vanishes  for  two  points  of  the  form  {rp  +  s,  1)  and  {rp'  +  s,  1), 
where  p  and  p'  are  the  roots  of  (5)  and  r,  s  are  integers,  r  +  0,  and  conversely. 


38  THE   M.^DISON   COLLOQUIUM. 

We  are  now  ready  to  prove  that  any  rational  integral  invariant 
If  with  integral  coefficients,  of  the  group  G  is  a  rational  integral 
function  of  L  and  Q  with  integral  coefficients. 

After  removing  possible  factors  L  and  Q,  we  may  assume  that 
I  vanishes  for  no  special  point.  If  /  is  not  a  constant,  it  vanishes 
at  a  point  (c,  d)  and  hence  at  the  w  distinct  points  conjugate  with 
(c,  d)  under  the  group  G.     The  invariants* 

P±}  JiCp— 1) 

(11)  q=Q'  ,     l  =  L    ' 

are  of  degree  co.     The  constant  r,  determined  by 

q{c,  d)  +  T  '  l{c,  d)  =  0         (mod  p), 

is  a  root  of  a  congruence  of  a  certain  degree  t  with  integral  coef- 
ficients and  irreducible  modulo  p.  Now  g  +  tHs  a  factor  of  I. 
Since  q,  I  and  I  have  integral  coefficients,  I  has  also  the  factors 

(12)  q-^rH,     q  +  T^%      -",     q  +  r^'-'l 
For,  by  Galois's  theorem  mentioned  above, 

are  the  roots  of  our  irreducible  congruence  of  degree  t.  Since 
the  conditions  which  imply  that  9  +  zl  shall  be  a  factor  of  I  are 
congruences  satisfied  when  2  =  r,  they  are  satisfied  when  z  =  r^*. 
Hence  if  we  multiply  q  -\-  tI  hy  the  product  of  the  invariants 
(12),  we  obtain  an  invariant  T  with  integral  coefficients  modulo  p. 
Since  L  and  Q  have  no  common  factor,  no  two  of  the  functions 
q-\-  rl  and  (12)  have  a  common  factor.  Hence  7"  is  a  factor  of  I. 
Proceeding  in  like  manner  with  I/T,  we  arrive  finally  at  the 
truth  of  the  theorem.f 

3.  Invariants  of  Smaller  Binary  Groups. — We  shall  later  need  the 
theorem  that  a  fundamental  system  of  rational  integral  invariants 

*  If  p  =  2,  we  omit  the  divisor  2  in  the  exponents. 

t  Proved  less  simply  .in  Transactions  of  the  American  Mathematical  Society, 
vol.  12  (1911),  p.  1.  Still  simpler  is  the  proof  that  various  coefficients  of  an 
invariant  are  zero,  Quarterly  Journal  of  Mathemalics,  1911,  p.  158. 


INVARIANTS  AND   NUMBER  THEORY. 


39 


of  the  group  composed  of  the  p  powers  of  the  transformation 
(13)  x'  =  x-\-  y,    y'  =  y        {mod  p) 

is  given  by  y  and  X,  where 


(14)     X  =  x{x-\-y){x-\-2y) •  •  •  {x-\-p—\y) ^x^—xy^^         {mod  p). 

Now  (1,  0)  is  the  only  special  point,  being  the  only  point 
unaltered  by  (13)  or  its  k\h  power,  k  <  p.  Hence  an  invariant 
not  having  a  factor  ?/  or  X  vanishes  at  imaginary  points  falling 
into  sets  each  of  p  points  conjugate  under  our  group.  As  at  the 
end  of  §  2,  the  invariant  is  a  product  of  factors  y^  +  rX  so 
related  that  the  product  equals  a  polynomial  in  y^  and  X  with 
integral  coefficients. 

Other  results  will  be  merely  stated,  since  they  are  not  pre- 
supposed in  what  follows.  Within  the  group  G  of  all  transforma- 
tions (1),  any  subgroup  of  order  a  multiple  of  p  is  conjugate 
with  one  containing  (13)  and  transformations  exclusively  of  the 
form 


(15) 


x'  ^tx-{-  ly,    y'  =  r^y        (mod  p), 


and  having  y  and  X  as  a  fundamental  system  of  invariants.* 
The  invariants  of  any  subgroup  whose  order  is  prime  to  p  have 
been  found.f 


4.  Invariants  of  the  Total  Group  on  m  Variables. — The  functions 


(16)     Lm  = 


Xi^" 
Xi^" 

Xx 


Pin 
T.     P 


i    Vws 


are  seen,  by  a  generalization  of  (8),  to  be  invariants  of  index  1 
and  0  respectively  of  the  group  Tm  of  all  linear  homogeneous 
transformations  on  a;i,  •  •  • ,  Xm  with  integral  coefficients  modulo  p. 

*  Bulletin  of  the  American  Mathematical  Society,  vol.  20  (1913),  pp.  132-4. 
t  American  Journal  of  Mathematics,  vol.  33  (1911),  p.  175. 


40  THE   MADISON   COLLOQUIUM. 

Since  Lm  is  an  invariant  of  Vm  and  has  the  factor  X\,  it  follows 
from  an  examination  of  its  diagonal  term  that* 

TO      p—1 

(17)  im  =  n  2  fe  +  Ck+iXk+\  +  •  •  •  +  CmXm)         (mod  y), 

k=l  Ci=0 

in  which  occurs  one  of  each  set  of  proportional  linear  forms  modulo 
2?.  A  like  proof  shows  that  the  numerator  of  Qms  is  divisible  by 
each  of  the  linear  functions  (17)  and  hence  by  Lm>  modulo  p- 

Making  use  of  the  theorem  in  §  2, 1  have  proved  by  inductionf 
that  the  m  invariants  Lm,  Qmi,  •  - ',  Qmw-i  are  independent  and 
form  a  fundamental  system  of  rational  integral  invariants  of  r^. 

A  fundamental  system  of  invariants  of  the  group  of  all  modular 
linear  transformations  on  two  sets  of  two  cogredient  variables 
has  been  obtained  very  recently  by  Dr.  W.  C.  Krathwohl  in  his 
Chicago  dissertation. t 

Formal  Invariants  and  Seminvariants  of  Modular  Forms, 

§§  5-13 

5.  Formal  Modular  Invariants. — Consider  a  binary  form 

f(x,  y)  =  aQX"  +  aix'-hj  +  •  •  •  +  ary\ 

in  which  x,  y,ao,  •  •  • ,  ar  are  arbitrary  variables.  The  transforma- 
tion (7)  with  integral  coefficients,  whose  determinant  A  is  not 
divisible  by  the  prime  p,  replaces  /  by  a  form 

0(Z,  Y)  =  AoX^  +  AiX'-'Y  +  •  • .  H-  ArY\ 
in  which 

(18)  Aq  =  /(a,  c),     Ai  =  ra'-^bao  +  •  •  • ,     •  •  • ,     Ar  =  fib,  d). 

A  polynomial  P(ao,  •  •  • ,  a^)  with  integral  coefficients  is  called  a 
formal  invariant  modulo  p  of  index  X  of  /  under  the  transforma- 

*  E.  H.  Moore,  Bulletin  of  the  American  Mathematical  Society,  vol.  2  (1896), 
p.  189.  His  proofs  do  not  use  the  invariantive  property.  A  lilce  remark  is 
true  of  the  proof  that  the  product  (17),  in  the  case  a;m  =  1,  is  congruent  to  a 
determinant  of  order  w  —  1,  then  obviously  equal  to  Lm,  by  R.  Levavasseur, 
Memoires  de  I'AcadSmie  des  Sciences  de  Toulouse,  ser.  10,  vol.  3  (1903),  pp. 
39-48;  Comptes  Rendus,  135  (1902),  p.  949. 

t  Transactions  of  the  American  Mathematical  Society,  vol.  12  (1911),  p.  75. 

J  American  Journal  of  Mathematics,  October,  1914. 


INVARIANTS   AND   NUMBER  THEORY.  41 

tion  (7)  if 

(19)  P(Ao,  Ai,  ■•-,  Ar)  =  A^P(ao,  a,,  -  -  ■ ,  Or)         (mod  v), 

identically  as  to  ao,  •  •  •,  Or,  after  the  A's  have  been  replaced  by 
their  values  (18)  in  terms  of  the  a,.  If  P  is  invariant  modulo  p 
under  all  transformations  (7),  it  is  called  a  formal  invariant 
modulo  p  of  /. 

The  term  formal  is  here  used  in  connection  with  a  form  /  whose 
coefficients  are  arbitrary  variables  in  contrast  to  the  case,  treated 
in  the  earlier  Lectures,  in  which  the  coefficients  are  undeter- 
mined integers  taken  modulo  j).  In  the  latter  case,  (19)  neces- 
sarily becomes  an  identical  congruence  in  the  a's  only  after  the 
exponent  of  each  a  is  reduced  to  a  value  less  than  p  by  means 
of  Fermat's  theorem  a''^  ^  a  (mod  p). 

The  functions  (18)  are  linear  in  oo,  •  •  • ,  Gt.  It  is  customary  to 
say  that  relations  (18)  define  a  linear  transformation  on  ao,  •  •  • ,  Or 
which  is  induced  by  the  binary  transformation  (7).  Let  V  be 
the  group  of  all  of  the  transformations  (18)  induced  by  the  group 
of  all  of  the  binary  transformations  (7).  Making  no  further 
use  of  the  form  /,  we  may  state  the  above  problem  of  the  de- 
termination of  the  formal  invariants  of  /  in  the  following  terms. 
We  desire  a  fundamental  system  of  invariants  of  group  V.  This 
problem  is  of  the  type  proposed  in  §  1;  the  group  F  is  a  special 
group  of  order  a  multiple  of  p.  Here  and  below  the  term  in- 
variant is  restricted  to  rational  integral  functions  of  ooj  •  •  •  j  cfr. 

A  theory  of  formal  invariants  has  not  been  found.  For  no 
form  /  has  a  fundamental  system  of  formal  invariants  been 
published.  Some  light  is  thrown  upon  this  interesting  but 
difficult  problem  by  the  following  complete  treatment  of  a 
binary  quadratic  form,  first  for  the  exceptional  case  p  =  2  and 
next  for  the  case  p  >  2,  and  preliminary  treatment  of  a  binary 
cubic  form. 

6.  Formal  Invariants  Modulo  2  of  a  Binary  Quadratic  Form. — 
Write 

(20)  S=ax'+  hxy  +  cy\ 


42  THE   MADISON   COLLOQUIUM. 

where  a,  h,  c  are  arbitrary  variables.     Under  the  transformation 

(21)  x  =  x'  +  y',     y  =  y', 
f  becomes  /',  in  which  the  coeflBcients  are 

(22)  a'=a,    h' ^  h,    c' =  a  +  h  ^- c        (mod  2). 

By  §  3,  the  only  invariants  under  d'  =  d,  c'  ^  c  -\-  d,  modulo  2, 
are  the  polynomials  in  d  and  c(c  +  d).  Take  d  =  a-\-  h. 
Hence  the  only  seminvariants  of  f  are  the  polynomials  in  a,  b  and 

(23)  s  =  c(c  +  a  +  b). 

Such  a  polynomial  is  an  invariant  of  /  if  and  only  if  it  is 
unaltered  by  the  substitution  (ac)  induced  by  (xy).    Thus 

(24)  b,    k  =  as,     q  =  b(a  ■}-  c)  +  a^  +  ac  -{•  c^  =  s  -\-  ab  -{-  a^ 

are  invariants  of  /.  Introducing  q  in  place  of  s,  we  see  that  any 
seminvariant  is  a  polynomial  in  a,  b,  q.  Consider  an  invariant 
of  this  type.  Since  its  terms  free  of  a  are  invariants,  the  sum 
of  its  terms  involving  a  is  an  invariant  with  the  factor  a  and 
hence  also  the  factors  c  and  a  -}-  b  -{-  c,  the  last  by  (22) .  Hence 
this  sum  has  the  factor  k,  and  its  quotient  by  k  is  an  invariant. 
By  induction  we  have  the  theorem: 

Any  rational  integral  formal  invariant  of  f  equals  a  rational 
integral  function*  of  b,  q,  k. 

7.  Formal  Seminvariants  of  a  Binary  Quadratic  Form  for  p>  2. 
Write 

(25)  /  =  ax'  +  2bxy  +  cy\ 

where  a,  b,  c  are  arbitrary  variables.  Under  the  transformation 
(21),/  becomes/',  whose  coefficients  are 

(26)  a'  =  a,     b'  =  a+b,     c'  =  a  +  2b  +  c. 


*  Replace  Xi,  Xz,  Xz,  of  §  4  by  a,  b,  c ;  then 

L,  =  bk{k  +  bq),     Qzi  =  h*  +  bk  -\-  q\     Qzi  =  6Y  +  ^'2^  +  h^k  +  k\ 


IXVARIANTS  AND  NUMBER  THEORY.  43 

Evident  formal  seminvariants  are  a,  A  =  6^  —  ac,  and 

p-i 

(27)  ^  =^  n  (ta  +  b)  ^  bP  -  ba^^         (mod  p), 

(28)  T.  =  n  {(f  -  k)a  +  2tb  +  c]      {k  =  0,  1,  •  . .,  p  -  1). 

Indeed,  the  linear  function  under  the  product  sign  in  (28)  is 
transformed  by  (26)  into  the  function  derived  from  it  by  re- 
placing ^  by  /  +  1.     As  in  (27), 

(29)  [Tt]a=o  =  cP  -  c6^i         (mod  p). 

Let  S{a,  b,  c)  be  a  homogeneous  rational  integral  seminvariant 
with  integral  coefficients.     Then,  by  (26), 

S{0,  b,  c)  =  S{0,  b,  26  +  c)         (mod  p). 

Thus,  by  §  3,  5(0,  b,  c)  equals  a  polynomial  in  b,  c^  —  cb'^^. 
Hence,  by  (29), 

S{a,  b,  c)  =  a(T{a,  b,  c)  +  4>{b,  jk)         (mod  p), 

where  a  and  <f)  are  polynomials  in  their  arguments.     Now 

62f  =  ^i  +  ai    ),     &p+2i  =  ^^i  j^  q(    ), 
Hence 

(30)  S  =  aX(a,  6,  c)  +  V'(iS,  A,  7/t)  +    Z   c^.-ft^i+i^.e.-^ 

where  X  and  i/'  are  polynomials  in  their  arguments,  and  di  is 
an  integer. 

When  y  is  multiplied  by  a  primitive  root  p  of  p,  a,  b,  c  are 
multiplied  by  1,  p,  p-,  respectively.  Hence  /3  is  multiplied  by  p, 
while,  by  (29),  jk  and  A  are  multiplied  by  p^.  If  therefore  we 
attribute  the  weights  0,  1,  2  to  a,  b,  c,  respectively,  and  the  weight 
5  +  2^  to  a''6V,  we  see  that  the  weight  of  every  term  of  7*  is 
congruent  to  2  modulo  p  —  1. 

We  can  now  prove  that  every  di  is  divisible  by  p.  For,  if  not, 
the  seminvariant  S  —  \p  has  a  term  of  odd  weight,  so  that  every 


44  THE  MADISON   COLLOQUIUM. 

term  of  X  is  of  odd  weight  and  hence  has  the  factor  b.     Thus 

S  —  xp  has  the  factor  b  and  therefore  the  factor  /3,  so  that  its 

terms  free  of  a  have  the  factor  b^.     But  this  is  impossible,  since 

2i  -{-  1  <  p  and  (29)  does  not  have  the  factor  b. 

Hence  S  —  \p  has  the  factor  a  and  the  quotient  is  a  semin- 

variant  of  the  form  aX'  +  xl/'.     Proceeding  in  this  way,  we  obtain 

the  theorem: 

Any  seminvariant  is  a  polynomial  in  a,  A,  /3  and  any  single  yk- 
Of  these,  /3  alone  is  of  odd  weight.     Hence  any  seminvariant  is 

a  polynomial  in  a,  A,  jk,  /5"  or  the  product  of  such  a  polynomial  by 

jS.     But 

p-i 

(31)  iS2  =  a^To  +  A(A  2    -  a^^y         (mod  p). 

To  prove  this,  it  suffices  to  show  that  the  second  member  is 
divisible  by  b  and  hence  by  /3,  and  being  of  even  weight  therefore 
by  /3-,  and  to  remark  that  each  member  of  (31)  reduces  to  6^^  for 
a  =  0.     Now 

[7o]6=o  =  n  if-a  +  c)  =  c\    n    (t-a  +  c) 

(=0  I        t  =  l  j 

p—l  p—\ 

=  c{c  '   —  (—  a)  ^  p        (mod  p)y 

p-\ 
aP[To]6=o  =  ac[{-  ac)  '    -  a^^^        (mod  p). 

But  A  reduces  to  —  ac  for  6=0.     Hence  the  second  member  of 

(31)  has  the  factor  b.     We  therefore  have  the  theorem: 

For  2?  >  2,  any  formal  seminvariant  of  a  binary  quadratic  form 
is  a  polynomial  in  a,  A,  70  or  the  product  of  such  a  polynomial  by  jS. 

8.  Formal  Invariants  of  a  Binary  Quadratic  Form  for  p  >  2. 
The  product 

(32)  r  =  JJ^  7/b    {k  ranging  over  the  quadratic  non-residues  of  p) 

is  an  absolute  invariant  of  /  under  the  group  G  of  all  binary 
transformations  with   integral  coefficients  taken  modulo  p  of 


INVARIANTS   ANT3   NUAIBER   THEORY.  45 

determinant  unity.     It  suffices  to  prove  that  this  seminvariant 
is  unaltered  bv  the  substitution 

(33)  a'  =  c,    c'  =  a,     h'  =  -  6,1 

induced  by  the  transformation  x  =  y',  y  =  —  x'.  ^  Under  (33), 
the  general  factor  in  (28)  is  replaced  by 

{t^-k){{T--K)a  +  2Th  +  c}, 


where 


7'  =  7o— ^,        K=       ^ 


f-k'  {f-kf 

Hence  K  is  quadratic  non-residue  of  'p  when  k  is.     Also, 

11(^2-^0= -^- In  {k-e)\  ^-kilH-ir-^-^k       (modp) 

t=Q  \      t=l  J 

if  A;  is  a  non-residue.  To  show^  that  the  product  of  the  resulting 
numbers  —  4^•  is  congruent  to  unity,  we  set  a:  =  0  in 

p-i 

(34)  n  {x  -k)  =  x'  +1         (mod  p), 

k 

and  note  that  2^-1  ^  1.     Hence  (32)  is  unaltered  by  (33)  and  is 
an  absolute  invariant  of  /  under  G. 
It  is  very  easy  to  verify  that 

(35)  J  =  070 

is  unaltered  by  (33),  so  that  J  is  an  invariant  of  /  under  G. 
If  an  invariant  has  the  factor  /3,  it  has  the  factor 

(36)  B  =  jSn7r     (/•  ranging  over  the  quadratic  residues  of  p). 

For,  under  the  substitution  (33),  64-ra  (r=#0)  becomes  T{c—h!r). 
By  choice  of  r,  we  reach  c  +  2ib,  where  t  is  any  assigned  integer 
not  divisible  by  p.     This  is  a  factor  of  7^  where  k  =  f. 

The  fact  that  B  is  an  invariant  may  be  verified  as  in  the  case 
of  (32)  or  deduced  from  the  fact  that 

a^  n  7/t  =  ayo  '  BT 


46  THE   MADISON   COLLOQUIUM. 

is  an  invariant,  being  the  product  of  all  non-proportional  linear 
functions  of  a,  b,  c  with  integral  coefficients  modulo  p. 

Hence  any  invariant  is  the  product  of  a  power  of  B  by  an 
invariant  which  is  a  polynomial  P  in  a,  A,  70. 

Since  jk  is  a  seminvariant  not  divisible  by  |S,  it  equals  a 
polynomial  in  a,  A,  70  (§  7).  But  if  a  =  0,  7^  =  70  (mod  p),  by 
(29),  and  A  =  6^  is  free  of  c,  so  that  7^  is  not  a  polynomial  in  a 
and  A  only.     Hence 

(37)  7;t  =  To  +  9k{a,  A)         (mod  p). 

For  2^  =  3,  the  polynomial  P  therefore  equals  a  polynomial  in 
a,  A,  72  =  r.  Now  an  invariant  4>{a,  A,  F)  differs  from  the 
invariant  0(0,  A,  F)  by  an  invariant  with  the  factor  a  and  hence 
the  factor  (35).  Treating  the  quotient  similarly,  we  ultimately 
obtain  the  following  theorem  for  the  case  p  =  3 : 

A  fundamental  system  of  formal  invariants  of  the  binary  quad- 
ratic form  f  modulo  p,  p  >  2,  is  given  by  the  discriminant  A  and 
F,  J,  B,  defined  by  (32),  (35),  (36).  The  product  of  the  last  three 
is  congruent  modulo  p  to  the  product  of  all  the  non-proportional 
linear  functions  of  the  coefficients  of  f. 

To  prove  the  theorem  for  2?  >  3,  note  first,  by  (37),  that  F, 
given  by  (32),  differs  from  70"^  by  a  polynomial  in  70,  a,  A  of 
degree  n  —  1  in  70,  where  n  =  {p  —  I)  12.  Hence  a  polynomial  in 
a,  A,  7o  equals  a  polynomial  in  a,  A,  70,  F  of  degree  at  most  n—  1 
in  7o.  Subtract  from  each  the  terms  of  the  latter  involving 
only  the  invariants  A,  F.  We  have  therefore  to  investigate 
invariants  of  the  type 

TO— 1  n—1 

(38)  Z  c^yo'P^(A,  F)  +  E  yo'(f>i{a,  A,  F), 

i=l  1=0 

in  which  the  Ci  are  integers,  while  Pi  and  0i  are  polynomials  in 
their  arguments,  and  <^t  has  the  factor  a.  If  every  Cf  =  0,  the 
invariant  has  the  factor  a  and  hence  the  factor  ayo  =  J,  and  the 
quotient  by  J  is  an  invariant  which  may  be  treated  similarly. 
The  theorem  will  therefore  follow  if  we  show  that  a  contradiction 


INVAEIANTS  AND  NUMBEH  THEORY.  47 

is  involved  in  the  assumption  that  a  certain  Cj  is  not  divisible  by  p. 
First,  the  remaining  d  are  divisible  by  p.  For  if  also  c,-  ^  0, 
let  Z-,A'"»T*»'  be  the  term  of  P,  of  highest  degree  in  A.  Since 
7o  and  T  are  of  degrees  p  and  np,  and  of  weights  =  2  and  0 
(mod  p  —  I),  yo^Pi  is  of  degree  pi  +  2ri  +  Sinp  and  of  weight 
=  2i  +  2r,-  (mod  p  —  1).     But  p  =  1  (mod  n).     Hence 

i+2ri  =  i+2ry,    22  +  2ri  =  2i  +  2ry        (mod  n), 

so  that  i  =  j  (mod  n).  But  i  and  j  are  positive  integers  <  n. 
Hence  i  =  j.  Multiplying  our  invariant  by  a  suitably  chosen 
integer,  we  have  the  invariant 


«-i 


(39)  7o^Py(A,  r)  +  Eyo'Ua,  A,  F),     Pj  =  AT«  +  •  •  -. 

t=0 

Now  —  (c  —  ka)h^^  is  the  term  of  highest  degree  in  h  in  yk- 
Hence 

(40)  70=  _c6p-i+  ...,     F  =  (r6"(P-i^+  •••, 

(41)  a  =  ni-  (c  -  ^-a))  =  (-  oY  +  (-  a)"         (mod  p), 

k 

where  k  ranges  over  the  non-residues  of  p,  the  last  following 
from  (34)  for  x  =  c/a.  Since  70  and  F  are  of  even  weights, 
only  even  powers  of  b  enter  (39).  Hence  an  invariant  (39)  is 
symmetrical  in  a  and  c.  We  shall  prove  that  this  is  not  the 
case  for  the  terms  of  highest  degree  in  6.     For  yo^Pj  this  term  is 

(42)  (-c)V6^    I3  =  j(p-l)  +  2r+sn(p-l), 

Let  da^W^T^*  be  one  of  the  terms  of  0i  in  which  the  exponent  of 
6  is  a  maximum.  Then  in  yo%i  the  highest  power  of  b  occurs 
in  the  terms 

(43)  da^^i-  cYaH'<,    /?.■  =  2/,-  +  goi{p  -  1)  +  Hv  -  D- 

Since  the  weight  and  degree  is  the  same  as  for  (42), 

22  +  /3i^2i+/3         (modp-1), 
(44) 

ei-\-  1+  QiU  +  jSi  =  ;  +  sn  +  /3. 


48  THE  MADISON   COLLOQUIUM. 

First,  let  (3i  =  /?.  Then  i  =  j,  ei=0  (mod  n),  whence  i  =  j. 
Thus  the  exponent  of  a  in  any  term  (42)  or  (43)  is  divisible  by  n, 
while  the  exponent  of  c  is  not,  being  congruent  to  j  modulo  n. 
Hence  the  coefficient  of  b^  in  the  sum  of  (42)  and  the  various 
terms  (43),  with  i  =  j,  is  not  symmetrical  in  a  and  c,  unless 
identically  zero.  But  (43)  has  the  factor  a  while  (42)  does  not. 
Hence  the  greatest  (Si  exceeds  jS. 

Next,  consider  a  set  of  terms  (43)  and  a  set  of  terms  of  like 
form  with  i  replaced  by  k,  all  being  of  equal  degree  in  b.  Then 
Pi  =  ^k.  By  (44i),  2i  +  0i^2k  +  Z?/,,  i  =  L  Consider  finally 
terms  (43)  with  jSf  constant.  In  them  the  residue  modulo  n 
of  ei  is  a  constant  =#  i.  For,  if  Ci  =  i,  then  2i  -{-  j3i  =  j  -\-  j3 
(mod  n)  by  (442),  so  that  j  =  0  (mod  n)  by  (44i).  Hence  these 
terms  (43)  are  not  symmetric  in  a  and  c  and  yet  do  not  cancel.* 

Our  fundamental  invariants  are  connected  by  a  syzygy;  for 
2?  =  3, 

(45)  B^  =  AT2  -I-  J(J  _  A2)2. 

9.  Formal  Invariants  of  a  Binary  Cubic  Form  for  p  4=  3, — 
We  have  seen  that  the  theory  of  formal  invariants  of  a  binary 
quadratic  form  is  dominated  by  the  invariantive  products  of 
linear  functions  of  the  coefficients.  While  these  products  de- 
pended upon  the  classification  of  integers  into  the  quadratic 
residues  and  the  non-residues  of  p,  we  shall  find  that  for  a  cubic 
form  it  is  a  question  not  merely  of  cubic  residues  and  "non-residues 
of  p,  but  of  the  larger  classes  of  reducible  and  irreducible  con- 
gruences.    Write 

/  =  ax^  +  dbx^y  -\-  Scxy^  +  dy^, 

thus  taking  p  =1=  3.     Under  transformation   (21),  /  becomes  /', 
whose  coefficients  are  given  by  (26)  and 

(46)  ^'  =  a  +  36  +  3c  +  ^. 


*  If  two  are  of  like  degree  in  c,  their  ^'s  are  equal  and  hence  their  /'s  are 
equal;  then,  if  of  like  degree  in  a,  their  e's  are  equal.  But  then  we  have  the 
same  term  of  0f. 


INVARIANTS   AND   NUMBER  THEORY.  49 

Hence  a,  /3  and  yic,  given  by  (27)  and  (28),  are  again  seminva- 
riants;  also, 

p-i 

(47)  5y,  =  !!{(<'-  3A-i  -  j)a  -\- W  -  k)b  +  Stc  +  d} 

(j,k  =  0,  ...,p-  1). 
Indeed,  if  Ft{a,  b,  c,  d)  is  the  function  in  brackets, 
Ftia',  h',  c',  d')  =  Fi+iia,  b,  c,  d). 
Any  invariant  with  the  factor  a  has  the  factor 

(48)  adoo  =  aU  {t\  +  2>fb  +  Ztc  -{- d)  =  /(I,  0)  n/(^,  1), 

whose  vanishing  is  the  condition  that  one  of  the  points  {x,  y) 
represented  by  /  =  0  shall  be  one  of  the  existing  p  +  1  real 
points  (1,  0),  {t,  1)  of  the  modular  line.  To  verify  algebraically 
that  the  seminvariant  (48)  is  an  invariant,*  note  that  it  is 
unaltered  modulo  p  by  the  substitution 

(49)  a'  =  -  d,    d'  ^  a,    b'  =  c,    c'  =  -  b, 

which  is  induced  on  the  coefficients  oi  f  by  x  =  y',  y  =  —  x'. 
The  product  P  of  the  8jk  in  ivhich  j  and  k  are  such  that 

\  =  t'-  m  -  j 

is  irreducible  modulo  p  is  a  formal  invariant. 

The  substitution  (49)  replaces  the  general  factor  of  (47)  by 

-  a  4- 3^6-  3(^2  _  i;)c  +  \d 

=  X{  (P  -  3KT  -  J)a  +  3(r  -  K)b  +  ZTc  +  d}, 


where 

f7  =  F  +  kf  +  tj, 
h=  -  2k^  +  QkH^  +  3ktj  +  t'j  +  f. 


*  For  any  form,  see  Transactions  of  the  American  Mathematical  Society, 
vol.  8  (1907),  pp.  207-208. 


50  THE  MADISON   COLLOQUIUM. 

We  are  to  show  that  there  is  no  integral  solution  x  of 

x^  -  ZKx  -  J=0        (mod  p). 
Multiply  this  by  X^  and  set  X.t  =  y.     Then 

y^  —  3gy  —  h  =  0        (mod  p). 
But  the  negative  of  the  left  member  is  the  result  of  substituting 

r  -\-  s  =  —  t,     rs  =  —  y  —  2k 
in  the  expansion  of  the  product 

The  latter  is  congruent  to  zero  modulo  p  for  no  values  of  r  and 
s  which  are  integers  or  the  roots  of  an  irreducible  quadratic 
congruence  with  the  integral  coejfficients  t,  —  y  —  2k. 

For  p  =  2,  P  =  5ii.     For  p  =  5,  P  is  the  product  of  two 
invariants* 

(50)  5ii522532541,       ^uhA^SA^AS, 

neither  of  which  is  a  product  of  invariants.  The  last  property 
is  true  also  of  the  following  invariants : 

7l5o3>       74^02,       T25o45l253o52o542> 

(51) 

The  product  of  these  seven  invariants  and  aSoo  equals  the  product 

of  all  the  linear  functions  of  a,  h,  c,  d,  not  proportional  modulo  5. 

For  p  =  2,  each  of  the  15  linear  functions  is  a  factor  of  just 

one  of  the  following  invariants  (no  one  with  an  invariant  factor) : 

(52)  adoo,     du,    fiyodou     K  =  b  +  c,     (a  +  b  -\-  c)5io. 
For  any  p  ^  3,  the  cubic  form  has  the  formal  invariant 

(53)  G  =  3(6cP  -  b^c)  -  (ad^  -  aH), 


*  In  those  linear  factors  of  the  first  which  lack  c,  the  product  of  the  coef- 
ficients of  a  and  6  is  a  quadratic  non-residue  of  5;  in  those  of  the  second  in- 
variant, a  quadratic  residue. 


INVARIANTS   AND   NUMBER  THEORY.  51 

and  an  absolute  formal  invariant*  K  of  degree  j)  —  1.  For 
p  =  5, 

(54)   X  =  6*  +  c^  -  h'-d?  -  a^(?  -  hcH  -  ahh  +  acd?  +  a'hd. 

Thus,  for  p  =  5,  K  and  the  discriminant  D  are  invariants  of 
degree  4,  and  weights  =  0,  2  (mod  4),  while  aSoo  and  G  are 
of  degree  6  and  weight  =  3  (mod  4).  It  follows  from  §  10  that 
there  are  no  further  invariants  of  degree  less  than  8.  Now  the 
first  and  second  invariants  (51)  are  of  degree  10  and  weight  =  1 
(mod  4).  Hence  if  either  is  expressible  as  a  polynomial  in  in- 
variants of  lower  degrees,  it  must  be  the  product  of  D  by  a 
linear  function  of  adoo  and  G.  This  is  seen  to  be  impossible 
either  by  a  consideration  of  the  terms  of  degree  ^  5  in  d  or  by 
noting  that  D  has  no  linear  factor.  Thus  7i5o3  or  745o2  occurs  in  a 
fundamental  system  of  invariants. 

Invariantive  products  of  linear  functions  of  the  coefficients 
of  the  cubic  form  therefore  play  an  important  role  in  the  theory 
of  its  formal  invariants.  Whether  or  not  they  play  as  dominant 
a  role  as  in  the  case  of  the  quadratic  form  is  not  discussed  here. 
We  shall  however  treat  more  completely  the  seminvariants. 

10.  Formal  Seminvariants  of  a  Binary  Cubic  for  p  >  3. — We 
shall  first  determine  the  character  of  the  function  to  which  any 
seminvariant  S{a,  b,  c,  d)  reduces  when  a  =  0.  Set  A  =  36, 
2B  =  3c,C  =  d.    Then  (26)  and  (46)  give 

A'  =  A,    B'  =  A  +  B,     C  =  A-\-2B  +  C         (when  a  =  0). 

Any  function  unaltered  by  this  transformation  is  (§  7)  a  poly- 
nomial in  A,  B^  —  AC,  70',  or  the  product  of  such  a  polynomial 
by  /S^  where  70'  and  /3'  are  the  functions  70  and  jS  written  in 
capitals.     But 

70'  =    n  m'b  +  StC  +d)  =  [5,o]a=0, 

t=0 


*  Transactions  of  the  American  Mathematical  Society,  vol.  8  (1907),  p.  221; 
vol.  10  (1909),  p.  164,  foot-note.  Bulletin  of  the  American  Mathematical 
Society,  vol.  14  (1908),  p.  316.    Cf.  Hurwitz,  I.  c. 


52  .        THE  MADISON   COLLOQUIUM. 

modulo  jp.     Hence 

(55)  S  =  aaifl,  h,  c,  d)  +  yk^4>(b,  q,  5yo)  (e  =  0  or  1), 
where  k,  j  may  be  given  any  assigned  integral  values  and 

(56)  ?  =  c2  -  ibd,     -  Sb'q  =  [D]a=o, 

D  being  the  discriminant  of/.    We  use  the  semin variants  (II,  §  2) 

(57)  S2=-b''  +  ac,     Ss  =  2¥-{-a{ad-Sbc). 

First,  let  p  =  5.  Then  q  =  c^  -\-  2bd.  We  have  the  formal 
semin  variants* 

(Tz  =  bq  —  a{ab  +  2cd), 

ai  =  K-  S.}=  q^+  a{abd  -  2ac2  +  b'^c  +  cd^), 

(75  =  bq^  +  a{-  ad^  -  bed"  +  ^cH  +  abc"  -  2¥c  +  a^b), 

a^=  q^  +  a {ad''  -  2bcd^  -  cW  +  abcH  -  2¥cd  +  a^bd-{-  2ac* 

(58)  -  b^c^  -  2aV  +  a¥)\ 
<n=  qyo+a{2 (62  -  flc)^^  _|_  ^2j^3  _  5^2(^3  _  2c4^2 _^  2aVd2 

-  2ac(62  -  ac)(^2  -  (^2  _  ac)W  -  2a'd^  +  2abc^d 

+  2a36cc^  +  2a¥c  +  3(6-  -  ac)c4  -  0452  _|_  2a-V}, 

while  2G  differs  from  670  by  a  multiple  of  a.  By  (55)-(58),  S 
differs  from  a  polynomial  in  the  seminvariants 

(59)  a,    D,    S2,    S3,    0-3,    K,    cTs,    ce,    o"!,    G,    70,     5oo 

by  a  function  aX  +  pbd^^  +  aqdl^,  in  which  p  and  cr  are  constants 
at  least  one  of  which  is  zero  (in  view  of  the  degree  of  the  terms) . 
But  the  increment  to  65y(,  under  transformation  (26),  (46),  is 

*  As  the  terms  with  the  factor  a  were  taken  all  of  the  proper  degree  and 
weight;  then  a  term  common  to  a  combination  of  the  seminvariants  (59)  was 
deleted.  Finally  the  coefficients  were  found  by  a  process  equivalent  to  the 
use  of  a  (non-linear)  annihilator,  Transactions  of  the  American  Mathematical 
Society,  vol.  8  (1907),  p.  205.  Expansions  were  made  in  powers  of  d  and  the 
terms  involving  d  rechecked.  As  each  remaining  term  involves  a  new  coef- 
ficient, there  is  no  doubt  as  to  the  existence  of  covariants  of  type  at,  ere,  aj, 
though  the  terms  free  of  d  were  not  rechecked. 


INVARIANTS   AND   NUMBER   THEORY.  53 

adlo  with  the  term  ad^^,  while  d  does  not  occur  to  this  power  in 
the  increment  to  a  function  X  of  degree  5^.  Again,  the  increment 
to  gSoo  has  the  term  2acZ^+^\  while  the  increment  to  a  function 
X  of  degree  5A  +  1  is  of  smaller  degree  in  d.  Hence  p  =  a  =  0. 
Then  in  aX,  X  is  a  seminvariant  which  may  be  treated  as  was  the 
initial  S. 

A  fundamental  system  of  formal  seminvariants  of  the  binary 
cubic  form  modulo  5  is  given  by  the  functions  (59). 

11.  For  p  =  2,  the  method  of  §  10  fails.  In  place  of  c  we 
now  introduce  the  seminvariant  K  =  b  -{-  c.  Then  the  trans- 
formation (26),  (46),  becomes 

(60)     a'  =  a,    K'  =  K,    b'  =  a  +  b,    d'  =  a  +  K  +  d. 

By  §3,  any  seminvariant  S{a,  K,  b,  d)  becomes  for  a  =  0  a 
polynomial  in  K,  b,  d{K -{-  d).  In  place  of  the  last  we  may 
use  5oo-     Hence 

S  =  acr  -\-  4){b,  K,  5oo),     5oo  =  d{a  -{-  K -\-  d). 

We  make  use  of  the  seminvariants 

A  =  ac?  +  6c  =  5oo  +  ^ai,    /3  =  6^  +  ab, 
(61) 

/5  +  A  =  6Z  +  a(6  +  ^). 

Hence  S  differs  from  a  polynomial  in  K,  5ooj  A,  jS  by  a  function 
ap  +  fer(/3,  Soo).  Let  (60)  replace  p  by  p'.  Then  p  -\-  p'  ^  t 
(mod  2).  Take  a  =  K  =  0;  then  (60)  is  the  identity  and 
0  =  T  (6^,  c?^)  identically  in  b,  d.  Hence  the  function  r  (jS,  5) 
is  identically  zero.  Thus  ap  and  hence  p  is  a  seminvariant. 
Hence  a,  K,  5oo,  A,  fi  form  a  fundamental  system  of  formal  semin- 
variants of  the  cubic  modulo  2. 

Note  that  A^  is  the  discriminant,  so  that  A  is  an  invariant. 
The  invariants  (52)  may  be  expressed  in  terms  of  our  semin- 
variants: 

6n  =  /  +  A,     iS7o5oi  =  ^(/3  +  /i^  +  aK)  (A  +  5oo), 
(62) 

(a  +  K)bi,  =  (a  +  K)  (a^  +  7)  =  aSoo  +  KI, 

where  7  =  cr  +  aK  +  5oo  is  an  invariant. 


54 


THE  MADISON   COLLOQUIUM. 


12.  Miss  Sanderson's  Theorem.* — Given  a  modular  invariant  i 
of  a  system  of  forms  under  any  modular  group  G,  we  can  con- 
struct a  formal  modular  invariant  I  of  the  system  of  forms  under 
G  such  that  I  =  i  (mod  p)  for  all  integral  values  of  the  coefficients 
of  the  forms.  As  the  proof  does  not  give  a  simple  method  of 
actually  constructing  /  from  i,  it  is  in  place  here  to  give  a  very 
interesting  illustration  of  the  theorem  with  independent  veri- 
fication. Take  as  i  the  fundamental  seminvariant  (—  l)^Pm-iam 
of  a  binary  form  /  (Lecture  II) .  Then  /  is  the  quotient  Lm+i/Lm, 
where  Lm  is  given  by  (16)  or  (17)  with  X:,  •  -  -fXm  replaced  by  the 
first  m  coefiicients  ao,  ai,  •  •  • ,  a^-i  of  the  binary  form  /.  Now 
X  =  x'  -{-  y',  y  =  y',  replaces  f{x,  y)  by  a  form  in  which  the 
coefficient  a/  is  a  linear  function  of  ao,  •  •  • ,  ay.  Hence  Lj  is  a 
formal  seminvariant  of  /  modulo  p.     First, 


ao 


«i 


-^  ao  =  ao^  ^ai  —  ai^ 


is  a  formal  seminvariant  which  reduces  to  —  Pofli  for  integral 
values  of  ao,  ai,  where  Po  =  1  —  ao'^^-     Compare  (27).     Next, 


ao^' 

ai^- 

a2^' 

ao^ 

fli^ 

a.f 

ao 

ai 

02 

L.= 


C  =  LzjLi  =  a2^"-  -  Qi^Q,  +  a^U-^^ 
where,  as  in  (10), 

j=0 


(mod  p), 


Q  = 


U 


=  S  ao^'-^^W' 


(s  =  p-  1). 


For  integral  values  of  the  a's,  we  have 

L2=0,    q  =  ao'  +  ai"  +  (p  -  l)ao'ax'  =  1  -  Pi, 

Pi=  (l-ao^)(l-ai^), 

modulo  p,  since  each  term  of  Q,  with  j  =#  0,  j  4=  Pt  is  congruent 
*  Transactions  of  the  American  Mathematical  Society,  vol.  14  (1913),  p.  490. 


INVARIANTS  AND  NUMBER  THEORY.  55 

to  ao*ai*-     Hence  C  =  Pia^,.    Similarly, 

UIU  =  -  az-P'  +  az-p"-Qz<2.  -  az'PQzx  +  03X3^^         (mod  p), 
where  the  Q's  are  defined  by  (16)  and  are  congruent  to* 

Qzi  =  Q(Lz/L2)^'  +  L^-^,    Qz2  =  {Lz/L^}^'  +  Q^, 
with  Q  as  above.     Hence  for  integral  values  of  the  a's, 
^31  =  (1  -  Pi)Pi«2^^  ^0,     Q32  =  1  -  Pi  (1  -  a2^')  =  1  -  P2, 

L^/Lz  ^  —  Pidz- 

13.  Modular  Covariants. — Extending  the  usual  definition  of  a 
covariant  of  an  algebraic  form/  to  the  case  in  which  the  group  is 
the  set  of  all  linear  transformations  with  integral  coefficients 
taken  modulo  p,  we  obtain  the  concepts  modular  covariants  or 
formal  modular  covariants  according  as  the  coefficients  of  / 
are  integers  taken  modulo  p  or  are  indeterminates.  The  contrast 
is  the  same  as  in  §  5.  The  universal  covariants  obtained  in  §  2 
and  §  4  do  not  involve  the  coefficients  of/  and  hence  are  formal 
covariants. 

I  have  recently  provedf  that  all  rational  integral  modular 
covariants  of  any  system  of  modular  forms  are  rational  integral 
functions  of  a  finite  number  of  these  covariants.  In  the  same  paper 
I  proved  that  a  fundamental  system  of  modular  covariants  of  the 
binary  quadratic  form  (25)  modulo  3  is  given  by  the  form  f  itself, 
its  discriminant  A,  the  universal  covariants  L  and  Q,  together  witht 

q  =  {a+  c)  {b-  +  ac  —  1),    /*  =  ax^  +  bx^y  +  bxy^  +  cy^, 

(63)  Ci  =  {a}b  -  &3)a;2  +  2(62  +  ^c)  (c  -  a)xy  +  {b'  -  bc-)y\ 

C2  =  (A  +  a^)x-  -  26(a  +  c)xy  +  (A  +  c^)y\ 

Here  f^  is  a  formal  covariant,  which  is  congruent  to  /  for  integral 

*  Transactions  of  the  American  Mathematical  Society,  vol.  12  (1911),  p,  77. 

t  Transactions  of  the  American  Mathematical  Society,  vol.  14  (1913),  pp. 
299-310.  The  extension  to  cogredient  sets  of  variables  has  since  been  made 
by  Professor  F.  B.  Wiley,  and  will  be  pubHshed  in  his  Chicago  dissertation. 

J  No  one  of  the  eight  is  a  rational  integral  function  of  the  remaining  seven 
even  in  the  case  of  integral  coefficients  a,  b,  c  taken  modulo  3. 


56  THE   MADISON   COLLOQUIUM. 

values  of  x,  y.  Also  Ca  and  (as  here  written)  Ci  are  formal 
covariants.  Note  that  —  g-  is  the  invariant  (42)  of  Lecture  II. 
When  q  is  made  homogeneous  by  replacing  —  a  —  c  by  —  a'  —  c^,  • 
we  obtain  the  formal  invariant  F  =  72,  given  by  (32),  The 
resulting  eight  formal  covariants  of  /  do  not  form  a  fundamental 
system  of  formal  covariants;  not  all  the  formal  invariants  are 
polynomials  in  A  and  F  (§  8).  No  instance  of  a  fundamental 
system  of  formal  covariants  has  yet  been  published. 

The  method  of  proof  will  be  here  illustrated  by  the  new  and 
simpler  case  of  a  binary  quadratic  form  (20)  with  integral  coef- 
ficients modulo  2.     By  §  6  any  invariant  of  /  is  a  polynomial  in 

(24')  h,     ahc,     q=  (&  +  1)  (a  +  c)  +  ac, 

to  which  the  formal  invariants  (24)  reduce  modulo  2.  Such  a 
polynomial  is  congruent  to  a  linear  function  of  these  three  and 
unity,  since 

hq  =  ahc         (mod  2), 

Further,  any  semin variant  is  a  polynomial  in  a,  b  and  q  (§6), 
and  hence  is  a  linear  function  of  1,  a,  h,  ah,  q,  ahc.     For, 

aq^  a  -^  ah  -\-  ahc         (mod  2). 

These  results  are  in  accord  with  those  obtained  otherwise  in  §  14 
of  Lecture  II.     We  shall  now  prove  the  following  theorem: 

Every  rational  integral  covariant  K  of  the  hinary  quadratic  form  f 
modulo  2  is  a  rational  integral  function  of  /,  its  invariants  h  and  q, 
the  universal  covariants 

Q  =  x^  +  xy  +  7/^     L=  x-y  +  xy-, 

and  the  linear  covariant 

Z  =  (a  +  h)x  +  (6  +  c)y,     P  =  f -\- hQ         (mod  2). 

The  leading  coefficient  »S  of  i^  is  a  seminvariant  and  hence  is 
of  the  form  /  +  ra  +  sab,  where  r  and  s  are  constants,  and  I 
is  an  invariant,  a  linear  combination  of  the  invariants  (24')  and 
unity. 


INVARIANTS  AND   NUMBER  THEORY.  57 

First,  let  K  be  of  even  order  2n.     Then 

Ki  =  K-  IQ""  -  rf"  -  sbf^ 

is  a  covariant  in  which  the  coefficient  of  o:^"  is  zero  and  hence 
has  the  factor  y.     Thus  Ki  has  the  factor  L  and  the  quotient  is 
a  covariant  of  order  2w  —  3  to  which  the  next  argument  applies. 
Next,  let  K  be  of  odd  order: 

After  subtracting  from  K  constant  multiples  of  IQ""  and  blQ"-, 
in  which  the  coefficients  of  a:^"''"^  are  a  -\-  b  and  ab  +  b,  re- 
spectively, we  may  assume  that  S  is  an  invariant.  After  also 
subtracting  from  K  a  constant  multiple  of  ILQ""'^,  where  7  is  a 
Hnear  combination  of  the  invariants  (24')  and  unity,  we  may 
assume  that  Si  =  (3ia  +  ^2C,  where  the  j8's  are  functions  of  b 
only.  Then  the  covariance  of  K  with  respect  to  the  trans- 
formation (21)  gives 

Sx''''''+Si'x'y+ . . .  ^i^=S.r'""-l-  iS+Si)x'"-'y+ ' ' .      (mod  2), 

where  Si  denotes  the  function  *Si  formed  for  the  new  coefficients 
(22).     Hence 

Si'-  Si  =  P2(a  +  b) 

must  equal  the  invariant  S.  Since  ^2b  is  a  function  of  the  in- 
variant b,  jSott  must  be  an  invariant,  so  that  ^2  =  0.  Thus 
S  =  0  and  K  has  the  factor  L  as  before.  Hence  the  theorem  is 
true  for  covariants  of  order  co  if  true  for  those  of  order  co  —  3. 
But  it  was  proved  true  for  those  of  order  zero. 

By  a  similar  method  I  obtain  the  following  theorem: 
A  fundamental  system  of  covariants  of  the  binary  quadratic  form 
f,  given  by  (20),  and  the  linear  form  X  =  aox-\-  aiy  modulo  2  is 
given  by  f,  X,  /, 

^1  =  (aa2  +  j)x  +  {cai  +  j)y, 

Q,  L  and  the  invariants  b,  q,  {ai  —  l)(a2  —  1)  dnd 
j  =  (a  +  b)ai  +  (6  +  c)a2. 


58  THE   MADISON   COLLOQITIUM. 

Since  Qi  and  a2  are  cogredient  with  x  and  y,  the  function  j 
obtained  from  the  co  variant  Z  of  /  is  an  invariant  of/  and  X. 

The  reverse  of  the  last  process  is  important.  If  we  adjoin  to  a 
system  of  binary  forms  in  the  variables  x'  and  y'  the  linear  form 
yx'  —  xy',  any  modular  invariant  of  the  enlarged  system,  formal 
as  to  X,  y,  is  a  modular  covariant  of  the  given  system  with  x' ,  y' 
replaced  by  x,  y.  The  theorem  of  §  12  therefore  proves  the 
existence  of  certain  formal  co  variants.* 

Applications  of  Invariants  of  a  Modular  Group,  §§  14,  15 

14.  Form  Problem  for  the  Total  Binary  Modular  Group  V. — 
This  group  is  composed  of  all  binary  linear  transformations  (7) 
with  integral  coefficients  taken  modulo  jp  whose  determinant  A 
is  not  divisible  by  p.     By  (8), 

(64)  L{x,y)^M{X,Y),     Q{x,  y)  ^  Q(X,  Y)        (mod  p), 

so  that  L^^^  and  Q  are  absolute  invariants  of  F.  Hence,  of  the 
functions  (1 1),  (?  is  invariant  under  F,  while  I  is  unaltered  by  certain 
transformations  and  changed  in  sign  by  others.  Thus  a  homo- 
geneous function  of  q  and  I  having  a  term  which  is  a  power  of  q 
is  a  relative  invariant  of  F  only  when  an  absolute  invariant. 
Hence  if  p  >  2,  it  involves  only  even  powers  of  /,  and  by  the 
homogeneity,  only  even  powers  of  q.  Hence  any  absolute  in' 
variant  of  T  is  a  product  of  powers  of  L^"^  and  Q  by  a  polynomial 
in  q"*,  P,  where  y  =  lifp=2,y  =  2ifp>2. 

In  particular,  L'p~^  and  Q  form  a  fundamental  system  of  absolute 
invariants  of  F.  The  so-called  form  problem  for  the  group  F 
requires  the  determination  of  all  pairs  of  values  of  the  variables 
X  and  y  for  which  L^^^  and  Q  are  congruent  modulo  p  to  assigned 
values  X  and  fx,  either  integers  or  imaginary  roots  of  congruences 
modulo  p.     We  have  therefore  to  solve  the  system  of  congruences 

(65)  {L{x,y)}^'^\    Q{x,y)^fx        (mod  p). 


*  After  these  lectures  were  delivered,  I  saw  a  manuscript  by  Professor  O. 
E.  Glenn,  containing  tables  of  formal  concomitants  for  forms  of  low  orders 
and  moduli  2  and  3.  He  employs  transvection  between  the  form  and  the 
covariant  L  of  §  2. 


INVAEIANTS  AND  NUMBER  THEORY. 


59 


First,  let  X  ^  0 

x^'    y^ 

0  = 

z'P     y^ 

X       y 

For  z  =  X  ov  z^  y,  we  have 


=  Lz^'  -  QLz^  +  L^z        (mod  p). 


Hence  x  and  y  are  roots  of 

(66)  F(z)  =  sP'  -  /X2P  +  X2  =  0         (mod  p). 

Having  no  double  root,  this  congruence  has  p"^  distinct  integral 
or  imaginary  roots.     These  roots  are 

(67)  eX-hfY      (e,/=0,  1,  ...,p-l), 

where  X  and  Y  are  particular  roots  linearly  independent  modulo 
p.     For, 

(68)  F{eX  +  fY)  =  eF{X)  +  fF{Y). 

Hence  any  pair  of  solutions  x,  y  of  (65)  is  of  the  form  (7),  where 
a,  '  •  -y  d  are  integers,  whose  determinant  A  is  not  divisible  by 
p,  in  view  of  (64i)  and  X  ^  0. 

Conversely,  if  A'  and  Y  are  fixed  linearly  independent  solutions 
of  (66),  any  pair  of  linear  functions  of  X  and  Y  with  integral 
coefficients,  whose  determinant  is  not  divisible  by  p,  gives  a 
solution  of  (65).  Indeed,  by  (68),  x  and  y  are  solutions  of  (66). 
From  the  two  resulting  identities,  we  eliminate  X  and  ju  in  turn 
and  get 

M  =  Q{x,  y),     {L{x,  y)]p  =  \L{x,  y). 

Since  X  and  Y  are  linearly  independent  modulo  p,  L{X,  Y)  is 
not  divisible  by  p  [cf.  (6)].  Thus  L{x,  y)  ^  0  by  (64).  Hence 
(65)  hold. 

Hence,  for  X  ^  0,  the  form  problem  has  been  reduced  to  the 
solution  of  congruence  (66).  The  latter  will  be  discussed  here 
in  the  simple  but  typical*  case  in  which  X  and  fx  are  integers. 
Now  the  problem  to  find  the  real  and  imaginary  roots  of  a  con- 

*  For  the  general  case,  see  Transactions  of  the  American  Mathematical 
Society,  vol.  12  (1911),  p.  87. 


60  THE   MADISON   COLLOQI'IOI. 

gruence  with  integral  coefficients  is  at  bottom  the  problem  to 
factor  it  into  irreducible  congruences  with  integral  coefficients. 
When  V  is  an  integer,  z^  —  vz  is  a  factor  of  (66)  if  and  only  if 
V  is  a  root  of  the  characteristic*  congruence 

(69)  v^-fxv  +  \=0        (modi^). 

Such  a  binomial  is  a  productf  of  binomials  z'^  —  8,  irreducible 
modulo  p,  whose  degree  d  is  the  exponent  to  which  the  integer  v 
belongs  modulo  p.  Since  2p  —  1  <  p^,  the  function  (66)  has 
an  irreducible  factor  0(2)  of  degree  D  >  1,  not  of  the  preceding 
type  z"^  —  d,  and  hence  with  a  root  r  such  that  r^/r  is  not  congruent 
to  an  integer.  Thus  every  root  of  (66)  is  of  the  form  Cir+C2r^, 
w^here  the  c's  are  integers.  The  irreducible  factors  of  (66)  are  of 
degree  D  except  those,  occurring  only  ichen  (69)  has  an  integral 
root,  of  the  form  z'^  —  8,  where  dis  a  divisor  of  D. 

To  find  D,  note  that  by  raising  (66)  to  the  powers  p,  p^,  •  •  • , 
we  can  express  z^'  as  a  linear  function  U  of  z^  and  z.  Now  D  is 
the  least  value  of  t  for  which  k  =  z.  But  the  coefficients  of  It 
are  the  elements  of  the  first  row  of  the  matrix  of  S^~^,  where 


S  = 


(ri) 


* 


Note  the  analogj'  of  (66)  with  the  Hnear  differential  equation 

ha^^ng  the  solution  z  =  e"  if  t;  is  a  root  oi  v^  —  fiv  +  X  =  0.  Also,  (68)  holds. 
Make  dz/dt  correspond  to  zp  and  hence  dh/dt^  to  {zp)p.  Thus  the  differential 
equation  corresponds  to  (66),  and  the  integral  z  =  e"'  (viz.,  dz/dt  =  vz)  to 
gP  =  vz. 

t  Let  f{z)  be  an  irreducible  factor  of  degree  d.     Its  roots  are 

r,     rP  =  vr,     r^   =  v^r,     •  •  -,     rP  ~   =  v^'h, 
where  v^  =  \,  v^  ^  \,  Q  <  I  <  d.     Thus  d  is  a  divisor  of  p  —  1.     Hence 

2P-1  —  V  =  zP-'^  —  rP-^ 
has  the  factor  z'^  —  r''.     The  latter  has  a  root  r  in  common  with  /(z).     But 

(H)p-i  =  v"^  =  \. 
Thus  5  =  r''  is  an  integer.     Hence  f{z)  =  z''  —  8. 


im^ARIANTS   AND   NUMBER  THEORY.  61 

But  Id  =  z  implies  that  Id+i  =  z^.  The  condition  for  the  latter  is 
therefore  S^  =  1.  Hence  D  is  the  period  of  S.  But  (69)  is 
the  characteristic  determinant  of  S.  According  as  it  has  distinct 
roots  Vi  and  v^  or  equal  roots  v  =  ^fx  =  \^,  a  linear  substitution 
of  matrix  S  can  be  transformed  linearly  into  one  of  matrix* 


\0     vj'  \0    v)' 


According  as  the  characteristic  congruence  (69)  has  distinct  {real 
or  imaginary)  roots  or  a  double  root,  D  is  the  least  common  multiple 
of  the  exponents  to  ichich  the  distinct  roots  belong  modulo  p,  or  is  p 
times  the  exponent  to  which  the  double  root  belongs. 

Finally,   let   X  =  0.     By    (6),   either   y  =  0  or  x  —  ay  =  0 
(mod  p),  where  a  is  an  integer.     In  the  first  case, 

Q  =  xP~P,     x^"-  -  fxx^  =  0. 

If  /x  =  0,  then  x  =  y  =  0.  If  /i  4=  0,  the  roots  x  are  equal  in 
sets  of  p  and  hence  are  cxi  (c  =  0,  1,  •  •  •,  p  —  1),  where  xi  is  a 
particular  root  not  divisible  by  p.  In  the  second  case  x  —  ay^O, 
we  take  a;  —  a?/  as  a  new  variable  X  and  conclude  from  the 
absolute  invariance  of  Q  that 

Qix,  y)  =  Q{0,  y)  =  2/^'-^ 

We  thus  have  the  first  case  with  y  in  place  of  x. 

Using  similar  methods,  I  have  solved  the  form  problem  for 
the  total  group  of  modular  linear  transformations  on  m  variables. f 

15.  Invariantive  Classification  of  Forms. — ^Let 

(70)  0(.T,  2/)  =  a:«+  •••  (m  >  1) 

be  a  binary  form  irreducible  modulo  p  and  having  unity  as  the 
coefficient  of  the  highest  power  of  x.  Let  G  be  the  group  of  all 
modular  binary  linear  transformations  (1)  with  integral  coef- 

*  In  the  second  case  we  use  the  new  variables  x  and  x  —  vy. 
t  Transactions  of  the  American  Mathematical  Society,  vol.  12  (1911),  pp. 
84-92. 


62  THE   MADISON   COLLOQUIUM. 

ficients  of  determinant  unity.  Let  (f)i  =  4),  4>2,  •  •  • ,  4)k  denote 
all  the  forms  of  type  (70)  which  can  be  transformed  into  constant 
multiples  of  ^  by  transformations  of  G.  Evidently  their  product 
P  =  0102  •  •  •  4>k  is  transformed  into  CtP  by  any  transformation 
t  of  G.  The  constant  Ct  is  easily  seen*  to  be  congruent  to  unity. 
Hence  P  is  an  absolute  invariant  of  G.  If  m  >  2,  no  0;  vanishes 
for  a  special  point.  We  now  apply  the  theorem  in  the  first  part 
of  §  14.  Hence,  if  m  >  2,  the  absolute  invariant  P  is  an  integral 
function  with  integral  coefficients  of  the  invariants  q,  I,  each  ex- 
ponent of  q  and  I  being  even  if  p  >  2.  In  view  of  the  definition  of 
the  (f)i,  this  function  of  q  and  /  is  an  irreducible  function  of  those 
arguments  modulo  p. 

Two  binary  forms  shall  be  said  to  be  equivalent  if  and  only  if 
one  of  them  can  be  transformed  into  a  constant  multiple  of  the 
other  by  a  transformation  of  G.  A  set  of  all  forms  equivalent 
to  a  given  one  shall  be  called  a  genus.  Thus  0i,  - '  ■ ,  4>k  form  a 
genus.  All  of  the  irreducible  forms  (70)  separate  into  a  finite 
number/  of  distinct  genera;  let  Pi,  •  •  •,  P/  denote  the  products 
of  the  forms  in  the  respective  genera.  Thus  -Km  =  P\  •  •  •  Pf 
is  the  product  of  all  of  the  binary  forms  x^  -\-  •  •  •  irreducible 
modulo  p.  Hence  iTm  is  a  polynomial  in  q,  I  with  integral  coef- 
ficients. Hence  the  f  genera  of  irreducible  binary  forms  of  degree 
m  >  2  are  characterized  invariantively  by  the  f  irreducible  factors 
Pi(5'j  I)  of  Tmiq,  I)  modulo  p. 

We  shall  see  that  Xmiq,  I)  is  easily  computed.  By  finding  its 
factors  irreducible  modulo  p  in  the  arguments  q,  I,  we  shall  have 
invariantive  criteria  for  the  equivalence  of  two  irreducible 
binary  forms  of  degree  m.  For  example,  we  shall  prove  that 
TTz  =  q  —  I  ii  p  =  2,  so  that  all  irreducible  binary  cubic  forms 
modulo  2  are  equivalent.  Further,  tts  =  q"^  —  F  U  p  >  2,  so  that 
the  irreducible  cubic  factors  oi  q  —  I  are  all  equivalent,  also  those 
oi  q  -\-  I,  while  no  factor  of  the  former  is  equivalent  to  one  of  the 
latter. 


*  Transactions  of  the  American  Mathematical  Society,  vol.  12  (1911),  p.  3,  §  4. 
The  present  section  is  an  account  of  the  simpler  topics  there  treated  at  length. 


INVARIANTS   AND   NUMBER  THEORY.  63 

In  general,  let  m  be  a  product  of  powers  of  the  distinct  prime 
numbers  qi,  •  •  •,  q^,  and  set 

From  the  expression  for  7r,„  due  to  Galois  we  readily  obtain 

^      _  Fm   ■    UFmlqjgj   •    H/^m/g^g.gtgt   '  '  ' 

UFmiQi   '    UFmigigjg^   '  " 

in  which  the  first  product  in  the  numerator  extends  over  the 
2M(m  —  1)  combinations  of  gi,  •  •  •,  q^  two  at  a  time,  and  similarly 
for  the  remaining  products.  By  the  first  theorem  of  this  section, 
and  (11),  TTm  is  a  polynomial  in 

J^qy^qr^i^  K  =  p  =  L^ip-D  (7  =  1  if  2)  =  2,  7  =  2  if  p  >2). 
We  readily  verify  the  recursion  formula 

Ft  ^  QFU  -  KFt,         (mod  p), 
since  Fi  =  1,  F2  =  Q.     In  particular, 

Fz^J-K,    F,^Q(Fz^-KJ^'). 
Now  TTs  =  F3,  TTi  =  FilQ.    Hence 

T3^  J  -  K,    Ti  ^  Jp  -  Rp  -  KJp-^        (mod  p). 

The  first  of  these  results  was  discussed  above.  Next,  for 
p  =  2,  TTi  is  the  irreducible  quadratic  form  q-  —  P  —  Iq,  so  that 
all  quartic  forms  irreducible  modulo  2  are  equivalent.  For 
p  >  2,  7r4  vanishes  for  K  =  pJ,  where 

p^  =  1  —  p         (mod  p). 

Except  for  p  =  §,     p  is  a  quadratic  Galois  imaginary  since 

P^'  ^  1  —  pp  =  p         (mod  p). 

Thus  7r4  is  a  product  of  J  —  2iv  and  ^{p  —  1)  irreducible  quad- 
ratic forms  in  J,  K.  Some  of  the  latter  yield  a  quartic  in  q  and  I 
which  is  irreducible;  others  yield  a  quartic  which  is  a  product  of 
two  irreducible  quadratics  modulo  p.     A  simple  discussion  shows 


64  THE  MADISON   COLLOQUIUM. 

that  the  number  of  irreducible  factors  of  TTiiq,  l)  is  6^'  +  ^  +  1 
if  p  =  Sk  +  t  (t  =  ±  I  or  -  3),  but  is  6k  +  2  'd  p  =  8k -}-  3. 
We  have  therefore  the  number  /  of  genera  of  irreducible  quartics 
modulo  p.  For  quintics  and  septics,  the  analogous  discussion 
is  simple,  for  sextics  laborious. 

We  may  utilize  similarly  the  invariants  (16)  of  the  group  on 
m  variables,  obtain  expressions  in  terms  of  them  of  the  product 
of  all  forms  in  m  variables  of  specified  types  (as  quadratic  forms 
transformable  into  an  irreducible  binary  form,  non-vanishing 
ternary  forms,  non-degenerate  ternary  quadratic  forms,  etc.), 
and  hence  draw  conclusions  as  to  the  equivalence  of  forms  of  the 
specified  type.* 

*  Transactions  of  the  American  Mathematical  Society,  vol.  12  (1911),  pp. 
92-98. 


I,ECTURE  IV 

MODULAR    GEOMETRY    AND    COVARIANTIVE    THEORY    OF    A 
QUADRATIC   FORM   IN  m  VARIABLES   MODULO   2 

1.  Introduction. — The  modular  form  that  has  been  most  used 
in  geometry  and  the  theory  of  functions  is  the  quadratic  form 

(1)  qm  (x)  =  ^CijXiXj  +  ^hiX^     {i,j=  1,  •••,m;  i  <  j) 

with  integral  coefficients  taken  modulo  2.  In  accord  with 
Lecture  III,  we  shall  use  the  term  point  to  denote  a  set  of  m 
ordered  elements,  not  all  zero,  of  the  infinite  field  Fi  composed  of 
the  roots  of  all  congruences  modulo  2  with  integral  coefficients. 
We  shall  identify  such  a  point  {xi,  •  •  • ,  Xm)  with  {pxi,  •  •  • ,  pXm) 
where  p  is  any  element  not  zero  in  F^.  The  point  is  called  real 
if  the  ratios  of  the  a;'s  are  congruent  to  integers  modulo  2. 
Let  the  Cij  and  6,  in  (1)  be  elements  not  all  zero  of  the  field  F^. 
Then  the  aggregate  of  the  points  (.r)  =  (.Ti,  •  •  • ,  Xn^  for  which 
qjn{x)  =  0  (mod  2)  shall  be  called  a  quadric  locus,  in  particular, 
a  conic  if  to  =  3.  The  locus  is  thus  composed  of  an  infinitude 
of  points,  a  finite  number  of  which  are  real. 

While  our  results  are  purely  arithmetical,  we  shall  find  that 
the  employment  of  the  terminology  and  methods  of  analytic 
projective  geometry  is  of  great  help  in  the  investigation.  Usually 
the  proofs  are  given  initially  in  an  essentially  arithmetical  form. 
In  case  a  preliminary  argument  is  based  upon  geometrical 
intuition,  a  purely  algebraic  proof  is  given  later.  The  geometry 
brings  out  naturally  the  existence  of  a  linear  covariant,  which  is 
important  in  the  problem  of  the  determination  of  a  fundamental 
system  of  covariants. 

2.  The  Polar  Locus. — The  point  (/c?/i  +  Xzi,  •  •  • ,  Kijm  +  Xz^) 
is  on  q{x)  =  0  if 

(2)  K\{y)  +  K\P{y,  z)  +  Vg(z)  =  0         (mod  2), 
6  65 


66  THE   MADISON    COLLOQUIUM.     ' 

where 

(3)  Piy,  z)  =  ^Cij{yiZj  +  yjZ^)     {i,  j  =  1,  '•■,m;i<  j). 

If  (y)  is  a  fixed 'point,  all  points  (s)  for  which  P{y,  z)  =  0 
(mod  2)  are  said  to  form  the  polar*  locus  of  (y).  For  (z)  =  (y), 
each  summand  in  (3)  is  congruent  to  zero  modulo  2.  Hence  the 
polar  of  (y)  passes  through  (y).  If  (2)  is  on  the  polar  of  (y),  (2) 
has  a  double  root  k  :  X  and  the  line  joining  (y)  and  (2)  is  tangent 
to  ^  =  0. 

We  may  write  (3)  in  the  form 

(3')  P(y,  z)  =  uiyi  +  •  •  •  +  w^i/m, 

where 

■Ml  =  C12Z2  +  C13Z3  +  C1424  +    •  •  •   +  CimZm, 

U2  =  CnZl  +  C2323  +  C24Z4  +    •  •  •   +  C-Zm^, 

^^^  7/3  =   CuZi  +  C2322  +  C34Z4  +    •  ■  •   +  CsmZm, 

1(m  =   CimZi  +  C2m22  +  C3mZ3  +    '  '  *   +  Cm-lm^-l- 

There  is  a  striking  difference  between  the  cases  m  odd  and  m 
even. 

3.  Odd  Number  of  Variables;  Apex;  Linear  Tangential  Equation. 
Let  m  be  odd.  Then  the  determinant  of  the  coefiBcients  in  (4) 
is  congruent  modulo  2  to  a  skew-symmetric  determinant  of  odd 
order  and  hence  is  identically  congruent  to  zero.  Hence  we  can 
find  values  of  Zi,  •  •  • ,  Zm  not  all  congruent  to  zero  such  that 
Ui,  •  •  •  ,Um.  are  all  zero  modulo  2.  Thus  the  polars  of  all  points 
(y)  have  at  least  one  point  in  common. 

We  shall  limit  attention  to  the  case  in  which  the  pfaffians 

(5)     Ci=[23.--m],    C2=[134---m],     •••,    C„,=  [12  •  •  •  7/i-l] 

are  not  all  congruent  to  zero.     The  point  (Ci,  •  •  •,  Cm)  shall  be 

*  Take  *c  =  1  and  let  (2)  be  a  point  not  on  q{x)  =  0.  Then  (2)  is  a  quad- 
ratic congruence  in  X  with  coefficients  in  Ft  and  hence  has  two  roots  Xi  and  X2 
in  that  field.  Now  the  points  {y)  and  (2)  are  separated  harmonically  by 
{y  +  X12)  and  (y  +  X22)  if  and  only  if  Xi  s  —  X2,  that  is,  if  Xi  =  \z  (mod  2). 
But  the  condition  for  a  double  root  of  (2)  is  P  =  0  (mod  2). 


INVARIANTS  AND  NUMBER  THEORY.  67 

called  the  apex*  of  the  locus  q(x)  =  0.  Now  each  Ui  ^  0  if 
Zi  —  Ci,  • ' ' ,  Zm^  Cm.  Hence,  for  m  odd,  the  polars  of  all  points 
pass  through  the  apex. 

If  {y)  is  any  point  not  the  apex,  the  line  joining  {y)  to  the  apex 
is  tangent  to  q{x)  =  0  (§  2).  Thus  any  line  through  the  apex  is 
tangent  to  q{x)  =  0. 

For  TO  =  3,  it  is  true  conversely  that,  if  the  line 

(6)  XuiXi  =  0         (mod  2) 

is  tangent  to  q{x)  =  0,  it  passes  through  the  apex,  so  that 

(7)  K  =  XC^^i 

is  zero  modulo  2.  Taking,  for  example,  Uz  =1=  0,  we  obtain  by 
eliminating  Xz  from  (6)  and  q(x)  =  0  a  quadratic  equation  in 
Xi  and  X2  whose  left  member  is  the  square  of  a  linear  function 
modulo  2  if  and  only  if  the  coefficient  of  XiXi  is  congruent  to  zero. 
But  this  coefficient  is  the  product  of  /c  by  a  power  of  Uz.  Thus 
K  =  0  is  the  tangential  equation  of  q{x)  =  0. 

The  last  result  is  true  for  any  odd  m.  The  spread  (6)  is  said 
to  be  tangent  to  q{x)  =  0  if  the  locus  of  their  intersections  is 
degenerate.  Taking  «,„  4=  0,  and  eliminating  Xm,  between  (6) 
and  q{x)  =  0,  we  obtain  a  quadratic  form  whose  discriminant, 
defined  by  (24),  equals  a  product  of  k  by  a  power  of  Um,  and  hence 
is  degenerate  if  and  only  if  k  ^  0. 

We  thus  have  geometrical  evidence  that  k  is  a  formal  contra- 
variant  of  q(x),  i.  e.,  an  invariant  of  q(x)  and  HuiXi. 

To  give  an  algebraic  proof,  note  that  k  is  unaltered  when  xi 
and  Xj  are  interchanged,  while 

(8)  Xi  =   Xi    +  Xi,      X2  =   X2',       •  •  •,       Xm=   xJ 

replaces  q{x)  by  q'(x')  in  which  the  altered  coefficients  are 

(9)  h'z' =  b2  +  hi -{- cn,     Cu  =  Coi  +  cu    (i  =  3,  -  ■  • ,  m). 


*  After  these  lectures  were  delivered,  I  learned  that  Professor  U.  G. 
Mitchell  had  obtained,  independently  of  me,  the  notion  apex  ("  outside  point ") 
for  the  case  m  =  3,  Princeton  dissertation,  1910,  printed  privately,  1913. 


68 


THE    M.VDISON   COLLOQUIUM. 


The  pfaffians  Co,  •  •  • ,  Cm  are  unaltered  modulo  2,  while 

(10)      Ci'=Ci+C2,    W2'=W2+Wi,    Ui'=Ui    {i^2)        (mod  2). 

Hence  k  is  unaltered  modulo  2.     Note  that 


(11) 


^2  = 


0 

C\J.     IX. 

Cl2 

Cl3 

■  •       Cim 

Ui 

Cl2 

0 

•               • 

C23 

•                  • 

•  '       Com 

•                •                • 

U2 

• 

Clm 

C2m 

C3m       • 

••     0 

Wm 

Wl 

W2 

Us 

••       llm 

0 

(mod  2). 


"We  saw  that  Ci,  -  •  • ,  Cm  are  cogredient  with  Xi,  '  ",  ^m-  This 
is  evident  from  the  fact  that  the  apex  is  covariantively  related 
to  q{x).  Hence  if  we  substitute  Ci  for  Xi,  -  >  • ,  Cm  for  Xm  in  (1), 
we  obtain  the  formal  invariant 

(12)  qUC)  =  i:CijCiCj  +  i:biC^    (i,  j  =  l,  • .  • ,  m;  i  <  j). 

If  this  invariant  vanishes,  the  apex  is  on  the  locus,  which  is 
then  a  cone.  Indeed,  by  (2),  every  point  on  the  line  joining  (C) 
to  a  point  on  q{x)  =  0  lies  on  the  latter.  Hence  q{;x)  can  be 
transformed  into  a  form  in  7?i  —  1  variables  and  hence  has  the 
discriminant  zero.  To  argue  algebraically,  let  new  variables  be 
chosen  so  that  the  apex  becomes  (0,  •  •  • ,  0,  1).  The  polar  of  any 
point  (?/)  passes  through  the  apex.  Taking  Si  =  0,  •  •  • ,  Zm-i  =  0, 
Em  =  1  in  (4),  we  see  that  the  polar  (3')  becomes  CimVi  +  •  •  • 
+  Cm-\mym-\y  which  must  vanish  for  arbitrar}^  y's.  Hence 
hmXm  is  the  only  term  of  (1)  involving  Xm.  But  the  apex  is  on 
the  locus.  Hence  6m  =  0  and  q{x)  is  free  of  Xm.  The  converse 
is  obvious  from  (5). 

Whether  m  is  odd  or  even,  q{x)  has  the  invariant 

(13)  Am  =  n(co-  +  1)     (i,  j  =  I,  ••',m;i<  j). 

This  is  evidently  true  by  (9)  or  as  follows.  If  Am  =  1  (mod  2), 
every  Ca  =  0  and  q  =  (2fe,.T,)-;  while  if  Am  =  0,  at  least  one 
Cij  is  not  congruent  to  zero,  and  q  is  not  a  double  line. 

Hence  the  product  Amq(x)  is  a  co variant;  in  fact,  the  square 


INVARIANTS   AND   NUMBER  THEORY.  69 

of  the  linear  eovariant  Am^biXi.     We  shall  see  however  that  there 
exists  a  more  fundamental  linear  eovariant. 

4.  Covariant  Line  of  a  Conic. — Since  we  shall  later  treat  in 
detail  the  case  m  =  3,  we  shall  replace  (1)  by  the  simpler  notation 

(14)  F(x)  =  aiX2X3  +  a2XiX3  +  03^:12:2  +  biXi-  +  biXi'  +  632-32. 
Its  apex  is  (ai,  02,  as).     Its  discriminant  (12)  is 

(15)  A  =  F{ai,  02,  as)  =  aia^as  +  a-^bi  +  a^^^  +  az^bz. 
The  invariant  (13)  becomes 

(16)  A  =  aia^az  {(Xi  =  a,-  +  1). 

Consider  a  form  (14)  with  integral  coefficients  and  not  the 
square  of  a  linear  function.  Then  not  every  Oi  is  congruent  to 
zero  modulo  2.  By  an  interchange  of  variables  we  may  set 
as  =  1.     Replace  Xi  by  Xi  +  aiXz  and  X2  by  X2  +  02^:3.     We  get 

Z1Z2  +  6iA^2  +  62X2^  +  Axz\ 

Let  A  =  1.     Replace  xz  by  Xz  +  biXi  +  60X2.     We  get 

(17)  0  =  Z1Z2  +  AV. 

The  only  real  points  on  <^  =  0  (mod  2)  are  (1,  1,  1),  (1,  0,  0), 
(0, 1,  0).  In  addition  to  these  and  the  apex  (0,  0,  1),  the  only 
real  points  in  the  plane  are  (1,  1,  0),  (0,  1,  1),  (1,  0,  1).  These 
lie  on  the  straight  line 

(18)  Xi  +  Z2  +  Xs  =  0        (mod  2), 

Hence  with  every  non-degenerate  conic  modulo  2  is  associated 
covariantly  a  straight  line. 

The  inverse  of  the  transformation  used  above  is 

Xi  =  a!i  +  aiXz,     X2  =  a;2  +  a2Xz, 
Xz  =  biXi  +  62^:2  +  (1  +  oi&i  +  a2b2)xz. 
It  must  therefore  replace  0  by  the  general  form  (14)  having 


70  THE  iLU)ISON   COLLOQUIIBI. 

«3  =  A  =  1.     It  actually  replaces  (18)  by 

(6i  +  l).Ti  +  (62  +  l).'r2  +  (63  +  aiao  +  1).T3, 

in  which  we  have  added  A  +  1  =  0  to  the  initial  coefficient  of  Xz. 
Guided  by  symmetry,  we  restore  terms  which  become  zero  for 
03  =  1  and  get 

,,,,      X  =  Z  (/3.- +  1K-, 

(19)  .=1 

i3i  =  61  +  ociOis,     i32  =  62  +  aitts,     /^s  =  &3  +  (Xia2. 
Making  the  terms  homogeneous  we  obtain  the  formal  co- 
variant 

(20)  L  =  Bixx  4-  52.1-2  +  Bzxz, 

Bi  =  bi^  +  a2Ct3  +  a2^  +  as',     -B2  =  bo~  +  ctiOs  +  ai^  +  ai^, 
(21) 

-S3  =  &3^  +  01^2  +  ai^  +  a2^. 

Under  the  substitution  (aiaj)(bibj)  induced  upon  the  coefficients 
of  F  by  (xiXj),  we  see  that  Bi  and  Bj  are  interchanged.  Under 
(9),  viz., 

(22)  bo'  =  62  +  &i  +  03,     a/  =  ai  +  a2         (mod  2), 
there  results 

(23)  5i'  =  Bu    B2'  ^  B2  +  Bu     Bs'  ^  Bs     (mod  2). 

Hence  (20)  is  a  formal  covariant  of  F.  For  other  interpretations 
of  L  see  §  8. 

5.  Even  Number  of  Variables. — The  determinant  of  the  coef- 
ficients in  (4)  is  congruent  modulo  2  to  the  square  of  the  pfaffian 

(24)  A^  =  [123  •  •  •  m]. 

This  is  in  fact  the  discriminant  of  q-m,  which  is  degenerate  if  and 

only  if  A;„  =  0  (mod  2).     I  have  elsewhere*  discussed  at  length 

the  invariants  of  qm- 

*  Transactions  of  the  American  Mathematical  Society,  vol.  8  (1907),  p.  213 
(case  m  =2);  vol.  10  (1909),  pp.  133-149;  American  Journal  of  Mathematics, 
vol.  30  (1908),  p.  263;  Proceedings  of  the  London  Mathematical  Society,  (2), 
vol.  5  (1907),  p.  301. 


INVAEIANTS  AXD  NUMBER  THEORY. 


71 


If  Am  ^  0  (mod  2),  we  can  solve  equations  (4)  for  the  zs. 
Substituting  the  resulting  values  into  q{z),  we  obtain  the  tangen- 
tial equation  Um  =  0  of  q(x)  =  0.  For  m  =  2  and  m  =  4,  we 
get 

U2  =  C12U1U2  +  biiii^  +  6l2^2^ 
(25) 

U4.  =   [l234]2C34?'rW2+2(C23C24C34+&2cL+&3cL+&4C23)Wl^- 

Bordering  the  algebraic  discriminant  of  (1),  we  find  that 


(26)      2U^  ^ 


26i 

C12 

Cl3         • 

•  '        Clm 

Ui 

C12 

2bo 

C23 

•  •        Com 

Uo 

Clm 

Clm 

Czm        ' 

••        2hm 

Um 

Ui 

111 

II3 

••       Um 

0 

(mod  4). 


Finally,  let  A^  ^  0  (mod  2).  Then  all  of  the  first  minors  of 
the  matrix  of  the  coefficients  in  (4)  are  zero  modulo  2.  Hence 
the  polars  of  all  points  have  in  common  the  points  of  a  straight 
line  S.  Since  its  discriminant  vanishes,  q{x)  can  be  transformed 
linearly  into  a  quadratic  form  in  a:i,  •  •  • ,  Xm-i,  which  therefore 
represents  a  cone  with  the  vertex  (0,  •  •  •,  0,  1).  Let  (2)  be  the 
vertex  of  the  initial  cone  q{x)  =  0.  If  (x)  is  any  point  on  the 
cone,  (x  +  \z)  is  on  the  cone,  and,  by  (2),  P{x,  z)  is  congruent 
to  zero  identically'  in  .ti,  •••,  Xm-  Hence  the  linear  functions 
(4)  all  vanish.  Thus  the  line  S  meets  the  cone  in  its  vertex,  and 
Zm^  is  the  discriminant  of  qm~i{x),  while  Zi^  is  obtained  from  that 
discriminant  by  interchanging  m  and  ^.     For  example,  if  m=4, 

24^  =  C12C13C23  +  &1C23  +  &2C13  +  &3C12,  •  •  • , 

2r  =  C23C24C34  +  boch  +  63C24  +  64023- 

The  product  of  the  general  form  (1)  by  5  =  A;„  +  1  is  a  quad- 
ratic form  whose  discriminant  is  zero  modulo  2  and  hence  has 
the  vertex  (dzi,  •••,  dzm),  where  Zi^  has  the  value  just  given. 
Hence  8zi^,  •  •  • ,  dzm^  are  cogredient  with  Xi,  •  •  • ,  Xm- 

6.  Covariant  Plane  of  a  Degenerate  Quadric  Surface. — The 
product  of  g4  by  5  =  [1234]  +  1  is  a  quaternary  form  /  whose 


72  THE   MADISON   COLLOQUIUM. 

discriminant  is  zero  and  hence  can  be  transformed  into  a  form 
(14)  free  of  x^.  With  this  cone  F  =  0  is  associated  covariantively 
the  plane  /  =  0,  where  I  is  the  ternary  covariant  (19).  Hence/ 
has  a  hnear  covariant  L  which  reduces  to  I  when  64  =  0,  Cu  =  0 
(i  =  1,  2,  3).  Relying  upon  symmetry  and  the  presence  of  the 
factor  d,  we  are  led  to  conjecture  that 

L  =  5{6i  +  1  +  (Ci2  +  l)(Ci3  +  l)(Cu  +  1)1.^1  +   •  •  • 
^^'^^  +  5  { 64  +   1  +   (Ci4  +  1)  (C24  +   1)  (C34  +   1)  }  X,. 

It  is  readily  verified  algebraically  that  X  is  a  covariant  of  94. 

There  is  a  simple  interpretation  of  L.  If  [1234]  ^  0  (mod  2), 
then  5=0  and  L  is  identically  zero.  If  [1234]  =  0,  q^  is  de- 
generate and  can  be  transformed  into  (f)  =  a'l.ro  +  0*3^  or  a  form 
involving  only  xi  and  X2.  In  the  former  case,  L  =  xi-\-  X2  -{-  Xz. 
Of  the  15  real  points  in  space,  the  seven  (100.t),  (OIO.t),  (lll.r) 
and  (0001)  are  on  the  cone  0^0,  the  two  (OOLr)  are  on  the 
invariant  line  S  through  the  vertex  (0001)  of  the  cone  and  the 
apex  (0010)  of  the  conic  cut  out  by  0*4  =  0,  while  the  remaining 
six  (101a:),  (011a:),  (110a:)  lie  on  the  plane  L  =  0.  Hence  with  a 
degenerate  quadric  surface,  not  a  pair  of  planes,  is  associated 
covariantively  a  plane,  just  as  a  line  (19)  is  associated  with  a 
non-degenerate  conic  (14). 

Every  linear  covariant  is  of  the  form  IL,  where  I  is  an  in- 
variant. Every  quadratic  covariant  is  a  linear  combination  of 
the  lU  and  Iq^. 

7.  A  Configuration  Defined  by  the  Quinary  Surface. — A  q^ 
whose  discriminant  is  not  zero  modulo  2  can  be  transformed  into 

F  =  a:ia:2  +  a:3a:4  +  a^-. 

The  15  real  points  on  i^  =  0  (mod  2)  are  given  in  the  last  column 
of  the  table  below.  In  addition  to  these  and  the  apex  (00001) 
of  F,  there  are  just  15  real  points  in  space: 

1  =  (00011),  2=  (01001),   3=  (01011),    4=  (00101),    5=  (01101), 

6=  (00110),   7=  (OHIO),    8=  (10001),    9=  (10011),    a=  (10101), 

6=  (10110),    c=  (11000),   cZ=  (11010),    e=  (11100),    /=  (11111). 


INVARIANTS  AND  NUMBER  THEORY. 


'3 


These  lie  by  threes  in  exactly  20  straight  lines,  which  occur  in  the 
columns  of  the  table,  with  the  heading  "  Sides."  With  these 
lines  we  can  form  exactly  15  complete  quadrilaterals,  the  three 
diagonals  of  each  of  which  intersect*  in  a  point  on  F  ^  0,  given  in 
the  last  column.  The  columns,  with  the  heading  "Plane,"  give 
the  equations  defining  the  plane  of  the  quadrilateral.  In  each 
case,  the  two  equations  of  the  plane  have  in  common  with  F  ^  0 
a  single  real  point,  the  intersection  of  the  diagonals.  Thus  the  real 
points  on  i^  =  0  are  its  points  of  contact  with  these  tangent  planes. 


Sides 

Diagonals 

Plane 

Inter- 
section 

146 

157 

356 

347 

13 

45 

67 

xi=0,     X3+Xi-\-Xi  =  Q 

01000 

146 

lab 

496 

69a 

19 

4a 

66 

X-i=Q,       X3+X4  +  X5=0 

10000 

146 

W 

4d/ 

6de 

Id 

4e 

6/ 

Xi=Xi=X3-\-Xi+Xi 

11001 

157 

lab 

5ac 

76c 

Ic 

56 

(« 

Xi-\-X2+Xz=Xz-\-Xi-\-Xi=0 

11011 

157 

lef 

58e 

78/ 

18 

5/ 

7e 

X2=X3,       Xi=Xi+Xi-\rXt 

10010 

lab 

lef 

2ae 

26/ 

12 

af 

6e 

Xi=X3,       X2=Xz-\-Xi  +  Xi 

01010 

28c 

29d 

BSd 

39c 

23 

89 

cd 

X3=0,       Xi=X2-\-Xi 

00010 

28c 

2ae 

5ac 

58e 

25 

8a 

ce 

X4  =  0,       Xi=X2+Xi, 

00100 

28c 

26/ 

78/ 

76c 

27 

86 

cf 

X3=Xi,       Xl=X2  +  Xi-\-X5 

00111 

29d 

2ae 

69a 

6de 

26 

9e 

ad 

Xi=X3+Xi=X2+Xi 

01111 

29d 

26/ 

496 

4d/ 

24 

9/ 

bd 

Xi=Xi,       X2=X3+Xi+Xi 

01100 

347 

38d 

4d/ 

78/ 

3/ 

48 

Id 

X2=X4,      Xi=X3+Xi+Xi 

10100 

347 

39c 

496 

76c 

36 

4c 

79 

Xi=Xi+X2=X3+X5 

11101 

356 

38d 

58e 

&de 

3e 

5d 

68 

X2  =  X3  +Xi,       Xi  =  X2  +Xs 

10111 

356 

39c 

5ac 

69a 

3a 

59 

6c 

Xi,=Xi+X2  =  Xz  +  Xi 

11110 

8.  Certain  Formal  and  Modular  Covariants  of  a  Conic. — For 
conic  (14),  the  polar  form  is 


(28) 


Hence  if  two  sets  of  variables  yi  and  2;^  be  transformed  cogredi- 
ently  with  the  set  Xi,  this  polar  form  (28)  is  a  covariant  of  F 
and  the  two  points  (y),  (s),  in  an  extended  sense  of  the  term 

*  The  dual  of  the  theorem  of  Veblen  and  Bussey,  "  Finite  projective  ge- 
ometries," Transactions  of  the  American  Mathematical  Society,  vol.  7  (1906), 
p.  245. 


«1 

^2 

az 

2/1 

2/2 

2/3 

Sl 

22 

23 

74 


THE   M.\DISOX   COLLOQUIUM. 


covariant.  In  particular,  if  we  take  {y)  =  {x),  (z)  =  {p^),  we 
obtain  a  covariant  of  F  in  the  narrow  sense  used  in  these  lectures. 
In  particular, 


(29) 


K  = 


Xi 


02 

Xi 


fl3 

.^•3 


Xl-      X2^      .T3^ 


M  = 


ai 

a2 

as 

Xl 

X2 

Xs 

Xl' 

Xo' 

xz' 

are  formal  covariants  of  F.     While  the  discriminant  A,  given  by 
(15),  is  a  formal  invariant,  (16)  is  not.     But 


(30) 
(31) 


^  +  A  +  1  =  q;         (mod  2), 


a  being  a  formal  invariant  of  F.  By  (23),  the  B's  are  contra- 
gredient  to  the  x's  and  hence  to  the  a's,  so  that 

(32)  Ai  =  XaiBi  =  Xuihi^  +  "Eaia/  +  aiaaaa 
is  a  formal  invariant.     For  integral  values  of  ai,  hi, 

(33)  Ai  ^  A  =  2ai(/3v  +  1)         (mod  2). 

Any  form  with  undetermined  integral  coefficients  Ci,  C2,  •  •  • , 
taken  modulo  2,  has,  by  (21)  of  Lecture  I,  the  invariant 
(ci  +  1)(C2  +  1)  •  •  ■•  Thus  (16)  is  an  invariant  of  (7)  and  hence 
of  F.     Likewise  from  (19)  and  F  itself,  we  obtain  the  invariants 

(34)  J  =  /3i/52i33,     AJ  =  AYl{hi  +  1). 
In  (6)  we  made  use  geometrically  of 

(35)  X  =  W1.T1  +  U2X2  +  W3.T3. 

Now  F  +  ^X^  is  congruent  modulo  2  to  the  quadratic  form  derived 
from  F  by  replacing  each  hi  by  &,-  +  tu^.  Making  this  replace- 
ment in  A,  we  see  that  the  coefficient  of  t  is  congruent  to  k^,  where 

(36)  K  =    aitli  +   02^/2  +  03^3 

is  therefore  a  formal  invariant*  of  F  and  X.     Making  the  same 


*  Since  (36)  is  a  contravariant  of  F,  Zai(dC/dXi)  is  a  covariant  of  F  if  C  is. 
Taking  Q2,  Qi,  L  as  C,  we  get  K,  M,  A,  respectively. 


INVARIANTS  AND  NUMBER  THEORY.  75 

replacement  in  J  and  taking  t  and  ui  to  be  integers,  we  obtain 
as  the  coefficient  of  i  ^  ^^ 

(37) 

+  PlUlUz  +  PzUlUz   +  W1M2W3, 

a  modular  invariant  of  F  and  X.     By  the  theorem  used  above, 

(38)  U=    (2/1+    l)(i/2+    1)(«3+   1) 

is  an  invariant  of  X.  In  w  +  w  +  1,  we  replace  |Si  by  the  con- 
gruent value  -Si  +  1,  and  render  the  expression  homogeneous 
in  the  w's  and  5's  separately.     We  get 

(39)  CO  =  2(5i52  +  B{~  +  52^)^3-  +  ^B^hmz, 

a  formal  invariant  of  F,  X.  For,  it  is  unaltered  by  the  sub- 
stitution 

induced  by  {xiXj),  and  by  the  substitution  (23)  and  (10)  induced 
by  (8).     Let  the  coefficients  of  F  be  integers  not  all  even.     Then 
(39)  becomes 
(39')  2(^1/32  +  1)W3-  +  2(i3i  +  1)W2W3. 

Its  covariant  L  is  identically  zero.  Hence,  hy  the  table  in  §  9, 
if  CO  is  not  identically  zero  it  can  be  transformed  into  u^  -\-  u-r 
+  W1W2  and  hence  vanishes  for  a  single  set  of  integral  values  of 
Ml,  U2,  Uz.  These  are  seen  to  be  Ui  =  /3i  +  1.  Hence*  the  line 
L  =  0  is  the  only  line  with  integral  line  coordinates  on  the  line 
locals  (39). 

The  invariant  A  for  (39)  is  J  (its  discriminant  is  zero,  as  just 
seen).  Thus  a  knowledge  of  any  one  of  the  concomitants  L,  J, 
CO  implies  that  of  the  other  two. 

The  covariance  of  K  in  (29)  implies  that 


(40)     ^1  = 


X2        Xz 
X^      X^ 


^2  = 


Xi        Xz 
Xi^     Xz- 


•Tl 

Xi 

^3   = 

Xi' 

X2^ 

*  Also  thus:  just  as  the  point  conic  F  =  0  determines  its  hne  equation  (36) 
and  hence  its  apex  (a),  so  the  covariant  line  conic  (39)  determines  the  point 
equation  'LBC-Xi  =  0,  which  is  the  line  L  =  0  for  integral  values  of  the  coef- 
ficients. 


76 


THE   MADISON   COLLOQUIUM. 


are  contragredient  with  ai,  a^,  cts  and  hence  with  Xi,  X2,  X3,  and 

therefore  cogredient  with  ?<i,  Uz,  U3.     Thus  (39)  yields  the  formal 

covariant 

(410  W'  =  2(5i52  +  B,'  +  5.2^)^3'  +  i:Bi%h. 

From  this  or  (39'),  we  obtain  the  modular  covariant 

(41)  W  =  S(/3i^2  +  1)^3'  +  S(/3i  +  1)^2^3. 
In  these  notations  (29)  become 

(42)  K  =  i:ai^i,     M  =  2aiUx2''  +  X2Xs  +  X3''). 

Finally,  by  (16)  of  Lecture  III,  we  have  the  universal  co- 
variants 

Xi        X2        X3 


(43)     i3  = 


Xi       X2       Xz- 


Qi  =  ^x^W  +  S.ri%2a-3  +  x^x^xi, 

Q2  =  2.ri4  +  S.l'i2.T2-  +  XxX2X^Xx. 


x^     0:2"     .T3" 

The  covariant  line  L  =  0  of  a  non-degenerate  conic  i^  =  0  is 
determined  by  the  three  (collinear)  diagonal  points  of  the  complete 
quadrangle  having  as  its  vertices  the  apex  (a)  and  the  three 
intersections  of  i^  =  0  with  its  covariant  cubic  curve  iiC  ^  0. 

Fundamental  System  of  Covaeiants  of  the  Ternary  Form  F, 

§§  9-32 

9.  Invariants  of  F. — A  fundamental  system  of  invariants  of  F 
is  given  by  A,  A,  J.  It  suflfices  to  prove  that  they  completely 
characterize  the  classes  of  forms  F  under  the  group  of  all  ternary 
linear  transformations  with  integral  coeflBcients  modulo  2. 
This  is  evident  from  the  following  table 


Class 

A 

A 

J 

L 

a:ia:2  +  Xs^ 

1 

0 

0 

Xi  +  X2  +  0:3 

X1X2  +  .Ti^  +  Xo^ 

0 

0 

1 

0 

a:ia:2 

0 

0 

0 

Xi  +  X2 

Xi" 

0 

1 

0 

Xi 

0 

0 

1 

1 

0 

INVARIANTS  AND  NUMBER  THEORY.  77 

As  to  the  classes,  we  saw  in  §  4  that,  if  F  is  not  the  square  of  a 
linear  function  (i.  e.,  not  reducible  to  x-^  or  0),  it  can  be  trans- 
formed into  X1X2  +  hiXi  +  hiX^  -\-  A.T3-  and  hence  into  one  of 
the  first  tlu-ee  classes  of  the  table.     By  means  of  the  relations 

(44)  A^^O,    AJ^O,    A^^A,     A'^A,    r'=J        (mod  2), 
any  polynomial  in  A,  /I,  J  equals  a  linear  function  of 

(45)  1,    A,     A,    J,     AJ. 

These  are  linearly  independent  since  there  are  five  classes. 

10.  Leader  of  a  Comriant  of  F. — Let  8  be  the  coefficient  of  0:3" 
in  a  covariant  of  order  co  of  F.     Writing  (14)  in  the  form 

(46)  F=f-\-lxz-{-hzXz^,    f=hiXi^-{-azXiX2-\-h<xX'r,     l=a'2.Xx-]raxX2, 

we  see  that  the  leader  S  is  a  function  of  63  and  the  invariants  of 
the  pair  of  forms  /  and  I  under  the  linear  group  on  Xi,  X2. 

In  the  modular  co variants  forming  a  fundamental  system  for/ 
(§  13  of  Lecture  III),  we  replace  .Ti  by  ai  and  X2  by  02  and  obtain 
a  fundamental  system  of  modular  invariants  of  the  pair/  and  I: 

(47)  as,    cxia2,    </=&i62H-(&i+Ma3,    i=  (&i+a3)«i+(&2+a3)a2, 
where  ai  =  a,  +  1.     By  means  of  the  relations 

(48)  aiaij  =  0,     qj  =  j+  a^j         (mod  2), 

any  polynomial  in  the  four  functions  (47)  can  be  reduced  to  a 
linear  combination  of 

(49)  1,    az,    q,    a^q,    0:10:2,    0:10:203,    ocia2q,    0:10:2^35',    i,     azj' 

These  form  a  complete  set  of  linearly  independent*  invariants 

of/,  ^. 

*  Instead  of  verifying  as  usual  that  these  10  functions  are  linearly  inde- 
pendent, we  may  deduce  that  result  from  the  fact  that  there  are  10  classes : 

I  =  Xi,    f  =  03X1X2  +  013X2^     or     qxi'  +  asXiX'  +  azXr, 
I  =Q,     /  =  xr  +  X1X2  +  X2^,     X1X2,     xi-     or     0. 
Since  (47)  characterize  the  classes,  they  form  a  fundamental  system. 


78  THE   MADISON    COLLOQUIUM. 

Hence  <S  is  a  linear  combination  of  the  functions  (49)  and  their 
products  by  63.  Moreover,  S  must  remain  unaltered  modulo  2 
when  03  and  &i  are  replaced  by 

(50)  fls'  =  fla  +  «i,     &/  =  61  +  &3  +  ^2, 

which  are  the  only  altered  coefficients  of  the  form  obtained  from 
F  by  the  transformation 

(51)  .Ti  =  xi,    X2  =  X2,    .T3  =  X3'  +  xi         (mod  2). 
Both  requirements  are  evidently  met  by  the  functions 

(52)  1,     aia2,     63,     bsaiat 
and  any  invariant  of  F.     We  find  that 

A  =  aia2(as  +1),     A  =  aia^ch  +  j  +  ofa&s  +  ^3, 

(53)  J  =  aiazias  +  1)(63  +  1)  +  hj  +  a^hsj  +  ^3^  +  «iQ!2?, 
AJ  =  aia2(as+  1)(63 +!)(?+ 1). 

From  these  and  their  products  by  63,  we  see  that 

(54)  AJ,     h,J,    J,     63A,     hA,    A,    A 
contain  the  respective  terms 

bzaia2a3q,     hzaia2q,     aiaoq,     hzj,     hzaia2az,    j,    aia2az, 

while  no  one  involves  an  earlier  one  of  these  terms.  Hence  any 
linear  combination  of  the  functions  (49)  and  their  products  by 
63  is  a  linear  combination  of  the  functions  (52),  (54)  and 

(55)  03,     bzttz,     q,     hzq,     cizq,     bzUzq,     Ozj,     hzazj,     aia2azq. 
A  linear  combination  of  the  latter  is  of  the  form 

0-  =  mia3  +  m2q  +  rtizazq  +  m^az  j  +  maia2azq, 

where  mi,  •  •  •,  W4  are  linear  functions  of  63,  while  m  is  a  constant. 
The  coefficient  of  0361  is  seen  to  be 

p  =  ^2  +  W362  +  ^401  +  mh2ai{R  +  &3  +  1), 


INVARIANTS  AND  NUMBER  THEORY.  79 

where  i2  =  63  +  «2  is  the  increment  to  61  in  (50).     Set 

(56)  (r=pazbi-\-ra3+sbi+t  (p,  ■'-,  t  independent  of  az,  bi). 
Let  the  substitution  (50)  replace  a  by  a\     Then 

(57)  a'  —  (T  =  pRuz  +  pa-ibi  +  paiR  +  rai  +  sR. 
This  is  zero  for  every  as,  61  if  and  only  if 

(58)  pR  =  0,     pax  =  0,     rai  =  sR         (mod  2). 

For  p  =  m2  +  •  •  •,  pai  =  0  gives  mz  =  0,  ???4  =  7^2.  Then 
pR  =  0  gives  W2  =  0,  mhz  =  0,  whence  m=  0.  Thus  a  =  miaz, 
so  that  mi  ^  0.     Hence  the  leader  of  a  covariant  of  F  has  the  form 

(59)  I  +  bzh  +  caia2  +  f/oiiaofes, 
where  I  and  Ii  are  invariants,  c  and  d  are  constants. 

CovARiANTS  Whose  Leaders  are  Not  Zero,  §§  11-19 
11.  Consider  a  covariant  of  odd  order  co: 

(60)  C  =  Sxz-  +  Sixz'^-h'i  +  S^xz^^-W  +  •  •  • . 

If  S\  is  derived  from  *Si  by  the  substitution  (50),  then,  by  (51), 

(61)  Sx'  =  Sx  +  o^S  =  Sx  +  S         (mod  2). 

Give  <Si  the  notation  (56).  Then  S  is  given  by  (57)  and  has  no 
term  with  the  factor  0361.  Now  azbx  enters  no  term  of  (59) 
except  J  and  A  J  of  I  and*  63  J  of  63/1,  and  in  these  is  multiplied  by 

(62)  bzax-\-axa2,     «ia2(&2  +  1)(&3  +  1),     bzOixa2, 

respectively.  Since  the  latter  are  linearly  independent,  neither 
J  nor  AJ  occurs  in  the  I,  Ix  of  the  leader  (59).  Also,  A  and 
axoii  occur  only  in  the  combinations  ^  +  1,  axa^  +  1,  since  (57) 
has  no  constant  term.  The  coefHcients  of  .Ts"  in  L",  AL"^, 
{A  +  A)i'^  are  respectively 

(63)  63  +  ciici-i  +1,     Abz,    A  +  A63  -\-hzaxa2, 


*  AJ  is  not  retained  in  /i,  since  hzAJ  =  0,  AJ  being  (34). 


80  THE   MADISON   COLLOQUIUM. 

where  L  is  the  Hnear  co variant  (19).  After  subtracting  from  C 
a  Hnear  combination  of  these  three  covariants,  we  may  set 

S  =  mi(A  +  1)  +  W22A  +  vi^hs  +  mbsaia2. 

Since  ^s^sociai  =  0,  AJ  =  0,  the  leader  of  the  covariant  JC  is 

JS  =  miAJ  +  iriiJ  +  mzbzJ. 

Hence'  mi  =  ms  =  0.  The  coefficient  of  a^  in  S  is  now 
m2(aia2  +  63)  and  must  vanish  for  63  =  a^  since  it  is  of  the  form 
pR  by  (57).  Hence  7/I2  —  0,  Thus  S  =  mhzaia2.  For  co  >  1, 
mFL'^~'^  has  this  same  leader.     For  co  =  1, 

C  =  ???(63a:ic»;2a-3  +  h\(X2(XzXi  +  62aia3a'2)i 

which  satisfies  (61)  only  when  m  =  0.  Hence  every  linear 
covariant  is  a  linear  function  of  L,  AL,  AL;  every  covariant  of  odd 
order  co  >  1  differs  from  a  linear  combination  of  L",  AL"^,  AL", 
FL'^~^  hy  a  covariant  ichose  leader  is  zero. 

12.  In  the  covariants  of  order  47i 

(64)  /Q2",  IF-"",  U\  F-^-^U  (/ an  invariant), 
the  coefficients  of  Xz""^  are  respectively 

/,     hzl,     bz  +  aiao  +  1,     bz(Xia2. 

Linear  combinations  of  these  give  every  leader  (59).  Hence 
every  covariant  of  order  4n  differs  from  a  linear  combination  of  the 
covariants  (64)  by  a  covariant  whose  leader  is  zero. 

13.  In  the  covariants  of  order  co  =  4/i  +  2 

(65)  IQ2''F,  Q2"/-'-,  AQs^i^  {I  an  invariant), 
the  coefficients  of  0:3"  are  respectively 

bzL     bz  +  aia2  +1,     A  +  63(A  +  aia2az). 

The  sum  of  the  third  function  and  bz(A  +  A)  is  A  +  bzaia2' 
Hence  any  covariant  C  is  of  the  form  P  +  C,  where  P  is  a  linear 


IXVARIANTS  AND  NUMBER  THEORY.  81 

combination  of  the  co variants  (65),  while  C  is  a  covariant  whose 
leader  is  an  invariant.     For  co  =  2, 

C  =   SX3^  +  S1.r3.r1  +  &i2  +  X2({). 

This  is  transformed  by  (51)  into  a  function  having  Si  as  the 
coeflficient  of  Xi'.  Since  S  is  an  invariant,  *Si  =  S.  Thus  every 
coefficient  of  C  equals  <S.  Then  (51)  transforms  C  into  a 
function  in  which  the  coefficient  of  .ri'.r2'  is  zero,  so  that  *S  =  0. 
Hence  every  quadratic  covariant  is  a  linear  function  of 

(66)  F,    AF,    AF,    JF,    L\    M\ 

14.  There  remains  the  more  difficult  case  of  covariants  (60) 
of  order  co  =  4w  +  2  >  2.  If  5/  is  the  function  obtained  from 
iS,-  by  the  substitution  (50),  then 

(67)  5'/  =  Si,    S2'  =  S-\-Si^-  S2. 

Now  Si  is  unaltered  also  by  the  substitutions  (22)  and 

(68)  as'  =  as  +  Oo,     h.'  =  bo  +  63  +  01         (mod  2), 
induced  on  the  coefficients  of  F  by  the  transformations  (8)  and 

(69)  .Ti  =  .Ti',      .To  =  Xo',      X3  =  X3'  +  xV. 

15.  A  fundamental  system  of  invariants  of  F,  under  the  group  V 
generated  by  the  transformations  (8),  (51)  and  (69),  is  given  by 
A,  A,  J,  02,  63,  OiCk:2  and 

(70)  iS  =  61(63  +  a2). 

It  suffices  to  prove  that  these  seven  functions,  which  are 
evidently'  invariant  under  F,  completely  characterize  the  classes 
of  forms  F  under  F.     There  are  six  cases. 

(i)  63  =  a2  =  1-  Replacing  xx  by  .Ti  +  ai.T2  and  .T3  by 
X3  +  azX2,  we  get 

F  =  fixi'  +  A.r2'  +  .T3'  +  .ri.r3. 

(ii)  63  =  1,  00  =  0,  flia:2  =  1.  Replacing  Xs  by  .1-3  +  Os.i'i, 
we  get 

F  =  Aa-i2  +  62.1-2-  +  .rs-  +  x,xs. 
7 


82  THE   MADISON   COLLOQUIUM. 

If  A  =  0,  then  h%^  J.    If  A  =  1,  we  replace  xi  by  Xi  +  62-^2 
and  get 

a^i^  +  .^3^  +  .^2.^3. 

(iii)  &3  =  1,  a2  =  CLiOLi  ^  0.     Replacing  .1-3  by  Xz  +  hiXx  +  hiX2, 

we  get 

Xz'  +  Aa:i.T2. 

(iv)  63  =  0,  a2  =  1.  After  replacing  .T3  by  .^3  +  03.7:2,  we 
obtain  a  form  with  also  03  —  0.  Taking  this  as  F,  and  replacing 
a'l  by  .Ti  +  aiX2,  we  get 

hix-^  +  Aa:2^  +  2:1.^3. 

Replacing  xz  by  0-3  +  hiXi,  we  get  A.T2^  +  XiXz. 

(v)  hz  =  02  =  0,  010:2  =  1.     Replacing  .T3  by  .T3  +  03a;i  +  62-^2, 

we  get 

^x^  +  0:23:3. 

(vi)  63  =  02  =  010:2  =  0.  Then  F  is  the  binary  form/  in  (46). 
The  effective  part  of  F  is  now  the  subgroup  Ti  generated  by  (8). 
Now 

/3=&i,    ^+1  =  03,    J  =  5  +  (61  +  1)03,     5  =  62(6i  +  «3). 

These  seminvariants  hi,  az,  -B  of  /  completely  characterize  the 
classes  of  forms/  under  Fi.     For,  if  03  =  hi, 

f  =  hixi^  +  Bx2^  +  6i.Tia:2; 

while  if  03  =  5i  +  1,  we  replace  .Ti  by  Xi  +  620*2  and  get 

hiXi^  +  {hi  +  l).Ti.'r2. 

16.  The  number  of  classes  of  forms  F  in  the  respective  cases 
(i)-(vi)  is  4,  3,  2,  2,  6.  Hence  there  are  exactly  19  linearly 
independent  invariants  of  F  under  the  group  F.  As  these  we 
may  take 

1,     02,     Oia2,     ^>     hz,     hztto,     hzaia2,     hzA, 
A  =  6101  +  •  •  •,     02A  =  &iflia2  +  •  •  •, 
(71)  /3  =  61(63  +  02),     a.^  =  6163O0, 


INVARIANTS  AND  NUMBER  THEORY.  83 

(71)  Afi=bi(bz+1)A,    b3A=bib3ai-\ ,    a2bzA  =  bib3aiao-i , 

J  =  bxbibz  +  •  •  •,     a^J  =  bib^bza-i  +  •  •  •, 

bzJ  =  bibibziaiQi  +  oi  +  02)  +  •  •  •,     AJ  =  bib2bzA  +  •  •  • . 

These  are  linearly  independent  since  the  first  eight  do  not  involve 
61,  while  all  the  terms  with  the  factor  bi  in  the  next  seven  are 
given  explicitly,  likewise  all  with  the  factor  6162&3  in  the  last 
four.  Hence  the  19  functions  (71)  form  a  complete  set  of  linearly 
independent  invariants  of  F  under  the  group  T. 

17.  Hence,  in  §  14,  Si  is  a  linear  combination  of  the  functions 
(71).  By  (670),  S  +  Si  is  of  the  form  (57)  if  *S2  be  denoted  by 
(56).  Now  0361  occurs  in  J,  AJ,  bzJ,  aiJ,  AjS,  but  in  no  further 
function  (71).  In  the  first  three,  azbi  is  multipHed  by  the  linearly 
independent  functions  (62),  respectively;  in  the  last  two  by 
bzaia2  and  aiazibz  +1),  whose  sum  is  congruent  to  the  first 
function  (62).  Hence  the  part  of  *S  +  *Si  involving  J,  •  •  • ,  Aj3 
is  a  linear  combination  of 

(72)  (63  -\-  02)  J  =  bibobzaiao  +  bibzaiaoocz, 

(73)  J+bzJ  +  A^=  (63  +  l)(6i6oa:a2  +  M  +  A). 

But  61  occurs  in  just  six  of  the  functions  (71)  other  than  the 
five  just  considered.  Thus  the  factor  pai  of  61  in  (57)  is  a  linear 
combination  of  the  coefficients  of  bi  in  (72),  (73),  /3,  02/3,  A,  aoA, 
bzA,  a'UbzA.  Now  ai  is  a  factor  of  the  coefficients  of  61  in  all  except 
the  second;  third  and  fourth,  while  in  these  the  coefficients  are 

(63  +  I)b2aia2,     63  +  «2  +  1,     a2&3 

and  are  linearly  independent.  Hence  (73),  jS,  a2/3  do  not  occur 
in  *S  +  Si.  By  (57),  the  latter  has  no  constant  term  and  hence 
involves  1,  A  only  in  the  combination  ^  +  1.  This  cannot 
occur  since  the  total  coeSicient  of  az  must  be  of  the  form  jjR 
and  hence  vanish  for  63  =  02.  At  the  same  time  we  see  that 
the  sum  of  the  constant  multipliers  of  A,  a2A,  63A,  a263A  is  zero 
modulo  2.     Hence  *S  +  Si  is  a  linear  combination  of  the  functions 


84  THE   MADISON   COLLOQUIUM. 

«2,  bs,  hza2,  a\at,  and  the  last  six  in  (74)  below.  Like  (57),  this 
combination  must  vanish  for  a\  =  0,  63  =  a^.  Since  all  but  the 
first  three  of  the  ten  functions  then  vanish,  the  sum  of  the 
multipliers  of  these  three  must  be  zero  modulo  2.  Hence  S  -{-  S\ 
is  a  linear  combination  of 

&3  +  «2,     aiihz  +  1),     a\0i2,     ^aOiaa,     hzA, 
^'^^  Aa„    A(63+l),    A(ao63+l),     (bs  +  a^W. 

18.  Without  altering  the  invariant  S,  we  may  simplify  Si  by 
subtracting  from  C  constant  multiples  of  Z"*""^  K  and  its  product 
by  A,  where  K  is  given  by  (29),  and  hence  delete  a2{b$  +  1) 
and  A(ao63  +  1)  from  the  terms  (74)  of  Si.     Then 

Si  =  S  +  mAa-i  +  miA{b3  +  1)  +  vi2{b3  +  a2)J 

+  mzibs  +  f/o)  +  m^aia-i  +  mjjsaia-z  +  m^bzA. 

The  coefficient  T  of  a:3"~\i-2  in  C  is  obtained  from  Si  by  applying 
the  substitution  (0102)  (6162)  induced  by  (3:1X2).  In  view  of  the 
transformation  (8),  we  see  that  T'  =  T  -\-  Si,  where  T'  is  derived 
from  r  by  (22).     Hence 

S  =  (m  +  mi)A  +  mib^A  +  viobsJ 

+  (niA  +  m5&3)(aia2  +  ai  +  02)  +  W3&3  +  mobs  A. 

Let  S  be  the  sum  of  the  second  member  and  the  function  ob- 
tained from  it  by  the  substitution  (0203)  (&2^3)-  Thus  S  =  0. 
Taking  63  =  b^,  we  get  W4  =  W5  =  0.     Then 

2  =  (62  +  ^3)^,     /  =  '"lA  +  niiJ  +  ??i3  +  msA. 

Applying  to  S  the  substitution  (68),  we  get  (62  +  ai)7  =  0, 
Applying  (aia3)(6i63)  to  the  latter,  we  get  (62  +  ci^)!  =  0. 
Adding,  we  get  (ai  +  ch)!  =  0.  Applying  (50),  we  see  that 
asl  =  0.  Then  each  aj  =  0,  so  that  I  =  gA,  where  gf  is  a 
constant.  By  2  =  0,  </  =  0.  Thus  mx,  m^,  viz,  me  are  zei-o. 
Hence  S  =  wA,  Si  =  ??^Aa2.     But 

(75)         E  =  F{I/  +  AF)  +  (A  +  A)L^  =  A.rg^  -{ . 


INVARIANTS   AND   NUMBER   THEORY.  85 

Hence  C  —  Q'z"~^E  has  the  leader  zero.  A7iy  corariant  of  order 
CO  =  4?i  +  2  >  2  differs  from  a  linear  combination  of  the  co- 
variants  (65)  and  Qo'^~^E  by  a  covariant  whose  leader  is  zero. 

19.  Regular  and  Irregular  Covariants;  Rank. — A  covariant 
shall  be  called  regular  or  irregular  according  as  it  has  not  or 
has  the  factor  Lz,  given  by  (43).  The  quotient  of  an  irregular 
covariant  by  L3  is  a  covariant.  Hence  the  determination  of  all 
irregular  covariants  reduces  to  that  of  the  regular  covariants. 
If  a  covariant  has  a  linear  factor  it  has  as  a  factor  each  of  the 
seven  ternary  linear  functions  incongruent  modulo  2,  whose 
product  is  Lz.  Hence  a  regular  covariant  has  a  non- vanishing 
component  involving  only  xi,  .T3.  In  a  regular  covariant  C 
without  terms  .Tj"  (i.  e.,  with  leader  zero),  this  component  has 
the  factors  Xi,  Xz  and  (by  the  covariant  property)  also  .Ti  +  Xz. 
The  product  of  these  three  linear  factors  was  denoted  by  ^2 
in  (40),  Let  ^2"*  be  the  highest  power  of  ^2  which  is  a  factor  of 
the  component  and  let  n  be  the  degree  of  the  quotient  in  the  .r's. 
Then  C  may  be  given  the  notation 

3 

(76)  Rm,n=   ^fi^t^  +  XiX2Xz4), 

i=l 

where,  if  n  =  0,  /2  is  a  function  of  the  a's  and  6's  not  identically 
zero,  while,  if  n  >  0,  /2  is  a  function  also  of  Xi,  Xz  in  which  the 
coefficients  of  .ri"  and  .T3"  are  not  zero;  /i  is  a  function  of  x^,  Xz; 

fz  of  Xi,  X2. 

The  regular  covariant  (76)  shall  be  said  to  be  of  rank  m.  In 
an  irregular  covariant  the  component  free  of  X2  is  zero  and  hence 
is  divisible  bj^  an  arbitrary  power  of  ^2 ;  it  is  proper  and  convenient 
to  say  that  an  irregular  covariant  is  of  infinite  rank. 

Any  covariant  of  rank  zero  differs  from  one  of  rank  greater  than 
zero  by  a  polynomial  in  the  known  covariants 

(77)  A,    A,    J,    F,    L,    Q,. 

This  is  a  consequence  of  the  theorems  in  §§  11-18,  where  the 
polynomial  is  given  explicitly.     Any  product,  of  order  co  in  the 


86  THE  MADISON  COLLOQUIUM. 

x's,  of  powers  of  the  covariants  (77)  can  be  reduced  by  means  of 
the  syzygies 

JL  =  0,    AD  =  AF,    (A  +  ^  +  /  +  1)(FL  +  70  =  0, 

(78)  AK  =  0,    FU  +  (^  +  A)L'  +  AF  +  AQ2  =  LK, 

F'  +  Q,F  =  UK  +  (A  +  JW  +  (A  +  l)LG  +  (.4  +  l)Qi, 

to  a  sum  of  covariants  of  order  co  given  in  §§  11-18  and  a  linear 
function,  with  covariant  coefficients,  of  K,  Qi  and 

G=  Q2L+D  =  2t2[^3(^i  +  1).^/  +  (/3i|33  +  D.ts.ti 

(79)  +  ^1(^3  +  Dxi"]  +  xxX2X^m  +  /32  +  /33  +  1) 

X  (a;i.T2  +  a^i.Ts  +  X2X3)  +  S(/3i  +  l)a.'r]. 

Here  G  and  X,  given  by  (42),  are  of  rank  1,  while  Qi=  ^2^+iK2  (  ) 
is  of  rank  2.  As  this  theorem  is  not  presupposed  in  what  follows, 
its  proof  is  omitted.  However,  it  led  naturally  to  the  important 
relations  (75)  and  (79)  and  showed  that  no  new  combinations 
of  the  covariants  (77)  of  rank  zero  yield  covariants  of  rank  >  0, 
a  fact  used  as  a  guide  in  the  investigation  of  the  latter  covariants. 

Regular  Covariants  R,n.o,  §§  20-22 
20.  A  separate  treatment  is  necessary  for  covariants  (76) 
with  w  =  0.  Then  each  fi  is  a  function  of  the  coefficients  ay,  bj. 
Since  the  factor  ^3*"  of  the  part  /s^s"*  of  jR^o  free  of  X3  is  unaltered 
by  every  linear  transformation  on  Xi  and  X2,  /a  is  a  linear  com- 
bination of  the  functions  (49)  and  their  products  by  63.  Also, 
fs  must  be  unaltered  by 

(80)  Xi  =  xi  +  X3:    ai  =  ai  +  a^,     63'  =  63  +  &i  +  a?. 

Both  conditions  are  evidently  satisfied  by  the  ternary  invariants 
and  by  a^  and  q,  in  (47).     In  view  of  (53),  we  may  employ 

AJ,     J,     ttsA,     A,     asJ,     qA,     A 

to  replace  in  turn 

bzaia-ia^q,     b-iOiia-iaa,     azj,    j,     a-Jj^q,     aiazci^q,     aia-zas, 


INVARIANTS  AND  NUMBER  THEORY.  87 

since  a  term  previously  replaced  is  not  introduced  later.     Thus 
/s  is  a  linear  combination  of  these  seven  functions,  as,  q,  azQ,  and 

ai(X2,    Ciiaiq,     hz,     63O3,     hzq,     hzaia-i,     hzocia^q,     bzj,     bzazj. 

Give  to  any  linear  function  miaicxi  +  •  •  •  of  these  the  notation 

(T  =  aaibz  +  |Sai  +  763  +  5. 

Call  e  the  increment  61  +  ^2  to  63  in  (80)  and  employ  e  to  eliminate 
61.     Then  a  is  unaltered  by  (80)  if  and  only  if 

ae  =  0,    aaz  =  0,    fiaz  =  ye        (mod  2). 

Since  63  does  not  occur  in  q  or  j,  nor  ai  in  q,  we  have 

a  =  m&a2  +  7n7a2q  +  ms{e  +  02  +  az)  +  ^9^3(6  +  02  +  as). 

Thus  ae  =  0  gives  711^  =  m?  =  0,  mg  =  m^.  Then  aaa  =  0 
gives  TO9  =  0.     Now 

j3  =  mia:2  +  W2a2?,     7  =  Tiiz  +  m^az  +  Wsg, 

and  jSaa  =  7^  readily  gives  o-  =  0.  ^wy  function  of  63  anc?  ^Ae 
invariants  (49)  0/  /  anc?  /,  ?f  ^icA  Z5  unaltered  by  (80) ,  Z5  a  linear 
combination  of  the  ternary  invariants  (45)  and  az,  q,  azq,  asA, 
azJ,  qA. 

21.  For  n  =  0  and  w  even,  there  exists  a  co variant  (76)  in 
which  fz  is  any  function  specified  in  the  preceding  theorem. 
For,  if  /  is  any  ternary  invariant,  IQi^^^  has  fz  =  /.  By  (42) 
and  (41),  K""  and  W^''-  are  of  the  form  (76)  with  fz  =  az  and 
^1^2  +  1>  respectively;  they  may  be  multiplied  by  any  invariant. 
By  (19)  and  (47),  we  have 

(81)     i8iiS2  +  1  =  q  +  azA  +  A  +  1,     azq  =  azA  +  qA  +  OzJ. 

Hence  we  obtain  q,  then  qA,  qA,  and  therefore  azq.  Any  co- 
variant  with  n  =  0,  m  even,  differs  by  an  irregular  covariant  from  a 
linear  function  of 

IQr^hK^,hW^''    {I=l,A,A,J,AJ;Ii=l,A,J;h=l,A,A). 


88  THE   MADISON   COLLOQUIUM. 

22.  For  n=  0  and  m  odd,  we  may  delete  the  terms  cts/i  from 
fz  by  use  of  IiK"^.  First,  let  m  =  1  and  apply  transformation 
(51) ;  we  get 

R'=  U,'+M2    +   (/l  +  /3)s^3'  +   {X^'X^'XZ'  +  X^'\')<f>. 

Thus  ^  =  0.     Since  fz=l-\-  hf],  condition  /i  +  /a  =  /s'  gives 

I  =  hittibi  +  a2&2  +  0^363  +  aoaa  +  a  102). 

Add  to  this  the  relation  obtained  by  permuting  the  subscripts 
1,  2.     Thus 

0  =  hibi  +  &2  +  020:3  +  aiaz). 

The  increment  under  (22)  is  72(61  +  03  +  0203)  =  0.  Now  I2 
is  of  the  form  x  -\-  yA  -\-  zA,  where  x,  y,  z  are  constants.  From 
the  terms  in  61&2,  we  get  y  =  Q.  Then  x  =  z  =  0.  The  only 
covariants  are  therefore  IiK. 

Second,  let  m  >  1.     Then  XTF('»-i>/2  jg  ^f  ^he  form  (76)  with 
/s  =  tta?  +  o-z,  by  (81 1).     Hence  we  may  set 

fz  —  I  -\-  cq-\-  dqA  (c,  d  constants). 

In  R  given  by  (76),  let  g  denote  the  coefficient  of 

(83)  X1X2XZ  •  a-2"'a-32"'-l 

In  the  function  derived  from  R  by  the  transformation  (51),  the 
term  corresponding  to  (83)  has  the  coeflBcient  g  +  /i,  since  by 
(82)   the  ^i  parts  contribute  only  one  such   term,   that   from 

fi^r'%'-  Now 

fi=  I  +  cq'  +  dq'A,     q'  =  h.hz  +  (62  +  hz)ai. 

When  g  is  given  the  notation  (56),  g'  —  g  =  f\  is  the  function 
(57).  But  azhi  occurs  in/i  only  in  J  and  AJ  and  in  them  with 
the  linearly  independent  multipliers  (62).     Hence 

I  =  niiA  +  1)  +  ?i2A. 

The  coefficient  of  az  in  /i  is  now 

niaia2  +  712(^102  +  63)  +  dq'aiUo  =  p(bz  +  02). 


INVAEIANTS   AND   NUMBER  THEORY.  89 

Taking  &3  =  02,  we  see  that  ni  ^  n2  =  d  ^  0.  Thus  /i  =  cq'. 
By  (57)  for  ai  =  0,  63  —  02,  we  get  c  =  0.  Any  covariant  icith 
n  =  0  and  m  odd  differs  by  an  irregular  covariant  from  a  linear 
junction  of  K"",  AK*^,  JK"*  and,  if  m  >  1,  KW^"^-^^'-. 

CovARiANTS  OF  Rank  Unity,  §§  23-26 
23.  Henceforth  let  m  >  0,  n>  0  in  (76)  and  set 

(84)  /2  =  Sxs-  +  Six,--'xi  +  S2X,--W  +  •  •  •     (S  +  0). 

Since  S  is  unaltered  by  the  group  F  of  §  15,  it  is  a  linear  com- 
bination of  the  functions  (71).  We  may  omit  the  functions 
02(63  +  1)  and  Aa2(63  +1),  since  Z'"i'^  is  of  the  form  (76)  with 
S  =  02(63  +  1).     Thus 

(85)  5  =  7+  02/1  +  63/2  +  kia^a^  +  hh^aia^+h^+h^aifi+hA^, 

where  I  is  any  invariant,  7i  a  linear  function  of  1,  A,  J;  /a  one  of 
1,  A,  A,  J;  while  ^3  =  61(63  +  0:2). 

First,  let  m  =  1.  If  T  and  B  are  the  coefficients  of  0*2"  in  fz 
and  /i,  transformation  (51)  replaces  the  covariant  (76)  by  a 
function  in  which,  by  (82),  the  coefficient  of  Xix^'"^^^  is 

(86)  T+B=  T', 

where  T'  is  derived  from  T  by  the  induced  substitution  (50). 
But  T  is  obtained  from  <S  by  the  interchange  [23]  of  subscripts, 
and  B  from  T  by  [13].     We  thus  find  by  (86)  that 

I  =  hill  +  (^1  +  ^^262)  (ai  +  a^ai) 

+  A'3(ai6i  +  0262  +  0363  +  aiat  +  a2a3) 

+  kibiiaibi  +  0363  +  aia2  +  0203). 

Let  S  be  the  sum  of  the  second  member  and  the  function  obtained 
by  applying  (0203)  (6263)  to  it.     In  2  =  0,  set  62  =  63;  we  get 

{^*l  +   A'3  +  63(/t-2  +  Ica)  1  (02  +  03)"!   =    0,       A-3  =  kl,       ki  =  /i-2. 

Then  2  =  0  may  be  written  in  the  form 

(62  +  63)X  =  0,    X  =  /2  +  A-2(A  +  A  +  1). 


90  THE   M.\DISON   COLLOQUILTH. 

As  in  §  IS,  X  =  0.  Thus  I2  and  I  are  the  products  of  A  +  .4  +  1 
by  ^2,  ki,  so  that 

S  =  (ki  +  A'263)  (A  +  ^  +  1)  +  02/1  +  hibibs  +  biao  +  aiaa) 
(87) 

+  k2hi{a2hi  +  010:2)  +  k^Aibibs  +  61). 

For  n  odd,  S  is  the  increment  to  Si  under  (50)  and  hence  has 
no  term  containing  0361.  If  t  is  the  coefficient  of  J  in  7i,  0361 
occurs  in  (87)  only  in  iaiJ  and  in  the  final  part,  being  multiplied 
by  tttiaibz  and  koai<X2{bz  +  1),  respectively.  Hence  t  =  k5=  0. 
Since  S  is  of  the  form  (57),  the  coefficient  of  61  must  vanish  if 
ai  =  0.     Thus 

^'1(63  +  ^2)  +  ^•2&3a2  =  0,     A-i  =  ^2  =  0. 

Now  S  =  ail  I  =  02  (w  +  rA)  must  vanish  for  Oi  =0,  63  =  02 
by  (57);  then  A  =  00(62  +  03),  so  that  u  =  v  =  0,  S  =  0.  Any 
covariant  with  m  =  1  a7id  n  odd  differs  from  one  of  rank  >  1  by 
a  linear  function  of  KL^,  AKL"^. 

24.  For  m  =  1,  n  =  Av,  we  may  delete  aoli  from  (85)  by  use 

of  IiKQ2\    Set  /i  =  5a-2"  + h  ^nXs".     Then  (51)  replaces 

(76)  by 

R'  =    USXz"  +  SiXz^-'Xi  +   (Si  +  S2)Xz--W  +•••]  +  ^3/3 
+   (^1  +  ^3)[Bn{Xz-  +  XZ--W  +    •  •  •) 
+  Bn-lXiiXz^-"^  +  Xz^-^Xi  H )]  +    (.Tl.T2.T3  +  Xi^X^W. 

Since  Si  is  the  increment  of  ^2,  it  is  a  linear  combination  of  the 
functions  (74).  By  use  of  L^~^Qi,  L'^~^K^  and  their  products  by 
A  and  A,  we  may,  without  disturbing  S,  delete  from  Si 

63+q:iq;2+1,     Abz,    A+b^A+bzaiai,     02(^3+1),     02A(63+1). 

Hence  we  may  set 

'Si  =  tiibz  +  tti)  +  t-ibzaiOi2  +  ^3Aa2  +  ^4(63  +  a2)J. 

Applying  (0102)  (6160)  to  S  and  Si,  we  obtain  Bn  and  Bn-i. 
Let  I  be  the  coefficient  of  .Tj-Ts""^  in  0.     By  the  coefficient  of 


INVARIANTS  AND  NUMBER  THEORY.  91 

.Tia-2a-3  •  a-2.r3"~^  in  R',  we  have 

Bn^-Bn-i  +  l^V. 

For  I  given  by  (56),  Bn  +  Bn-i  is  given  by  (57).  By  the  coef- 
ficient of  ^361,  we  get  ^4  =  0.  The  coefficient  of  a^  must  vanish 
for  63  =  02.     Hence 

hoii  +  {h  +  ^3)ai«2  +  hoiiciih  =  0,     ki=  h^  0,     U  =  hy 

S  =  kobziA  +  ^  +  1  +  aittz  +  a2?'i). 

The  coefficient  of  ki  equals  that  of  ^23:3"'  in 

GFQ2'-'  +  AKL^  +  AKQi". 

Any  covariant  icith  m  =  1,  n  =  4v,  differs  from  one  with  m  ^  2 
by  a  linear  function  of  KL"",  AKL"",  IKQ^",  GFQ/-^  (/  =  1,  A,  J). 

25.  For  m  =  1,  n  =  4v  -\-  2,  we  may  delete  02/1  from  S, 
given  by  (87),  by  use  of  IiQ-^'M.  The  coefficient  of  ^2^:3''  in 
q^'G  is 

d  =  /33(/3i  +  1)  =  ^  +  (61  +  l)(aia2  +  63)  +  b^a^az. 

The  coefficient  of  ki  in  ^S  equals  d  +  QiA  +  02(63  +  1),  the  final 
term  of  which  was  reached  in  §  23,  and  02A  above.  The  coef- 
ficients of  k-a  and  k^  in  S  equal  Ad  and 

bibz{ax-\-ai)-\-bz{a2b2+aiaz-\-a2az+a2)=Ad-\-a2{J-\-\)-\-a2{bz-\-V), 

respectively.     Any  covariant  with   m  =  1,   n  =  Av  -{-  2,   differs 
from  one  with  m  ^2  by  a  linear  function  of  KL"^,  AKL"^,  IQ^G, 
hq.^M  (I  =  1,  A,  A;  h  =  1,  A,  J). 
For  use  in  §  26,  we  replace  Q2''M  by  Qi'FK,  noting  that 

(88)  M  =  {F-{-  U)K 

and  that  Q^UK  differs  from  KL^  by  a  covariant  of  rank  2. 

26.  By  the  last  four  theorems,  any  covariant  of  rank  1  differs 
from  one  of  rank  ^  2  by  CK  +  DG,  where  C  and  D  are  known 
covariants  of  rank  zero.     Taking  as  C\  and  Di  arbitrary  func- 


92  THE   AIADISON   COLLOQUIUM. 

tions  of  the  proper  degree  in  the  a;'s,  of  the  generators  (77)  of 
covariants  of  rank  zero,  I  found  the  syzygies  needed  to  reduce 
CiK-{-  DiG  to  an  expression  differing  from  the  above  CK  +  DG 
by  a  covariant  of  rank  ^  2,  in  which  those  of  rank  2  are  linear 
combinations  of  K^,  KG,  G^,  W,  Qi  and  the  new  one 

V  =  GF'  +  AQ.G  +  (A  +  ./  +  l)Q2FK 

^here  .  +  ^^'^"  +  ^^'^'  =  ''"'''"'  +  ' ' ' ' 

(90)  v=  02+  63(1  +  oiaz). 

The  only  new  syzygies  needed  for  this  reduction  are 

LG  =  QoL'  -{-  L^=  W,    FLK  =  AW  +  AQi  +  ( J  +  IW, 

(91)  (F  +  i^  +  Q2)K  =  (^  +  1)7.3, 

(A  +  1){FG  +  KL^  +  KQo)  +  JKQo  =  ALQi  +  C0Z3, 

in  which  co  is  an  invariant  not  computed.  Proof  need  not  be 
given  of  these  facts  since  we  presuppose  below  merely  the  ex- 
istence of  relation  (89)  which  may  be  verified  independently. 
Of  course,  the  fact  that  V  is  the  only  new  covariant  of  rank  2 
was  a  guide  in  the  later  investigation. 

CovARL\NTS  OF  EvEN  Raxk  lU  =  2/i  >  0,  §§  27-29 

27.  First,  let  n  be  odd.     In  the  covariant  (76)  replace  X3  by 
Xz  +  Xi.     In  view  of  (82),  we  get 

R'  =  fi'^'T  +  /3.^3'"  +  Uax'  +  ^^3')'^  +  (X,X2XS  +  Xr^X,)4>'. 

Using  the  notation  (84)  for  f^,  we  have  »Si'  =  Si-\-  S  in  f^'- 
Thus,  as  in  §  17,  *S  is  a  linear  combination  of  the  functions  (74). 
Now  Qi'^L"'  and  its  products  by  A  and  .4  +  A  are  covariants  (76) 
with  S  given  by  (63).  Using  also  A^*"i",  in  which  S  =  02(63+1), 
and  its  product  by  A,  we  may  set 

S  =  ^1(63  +  02)  +  /i2&3aia2  +  /i*3Aa2  +  ^^'iibs  +  ao)J. 

In  a:i.T2.T30,  let  g  be  the  coefficient  of 

a;ia-2a-3  •  Xi-^^'W'^'''^  =  {x2W)''x3''-^Xi. 


INVARIANTS   AND    NUMBER   THEORY.  93 

Such  a  term  occurs  in  neither  of  the  first  two  parts  of  R',  since 
they  are  functions  of  only  two  variables.  To  obtain  such  a 
term  from  the  third  part  of  R',  we  must  omit  terms  with  the 
factor  ^3-  (and  hence  .rr)  and  take  {x^x^Y'^  in  ^i"*^,  so  as  not  to 
make  the  degree  in  .T2  too  high.  Hence  if  T  be  the  coefficient  of 
Xz^  in  /i,  (/'=</+  T.  Now  (aia2) (&1&2)  replaces  S  by  T.  The 
resulting  T  must  be  of  the  form  (57).  By  the  coefficient  of 
O361,  ^-,1  =0;  cf.  (72).  By  the  coefficient  k^aibz  of  a^,  k^  =  0. 
Since  T  ^  0  for  Gi  =  0,  63  ^  a-z,  we  get  ^*i  =  ko.  Hence  S  =  kiv, 
where  ?jis  given  by  (90). 

For  n  =  1,  /2  =  8x3  +  SiXi.  Thus  Si  =  kiv',  where  v'  is 
derived  from  v  by  interchanging  the  subscripts  1  and  3.  Then 
Si'  =  Si  +  S  gives  ^-1  =  0. 

For  n  ^  3,  Qi'^~^L''~W  is  of  the  form  (76)  with  S  =  v,  since 
^sv  =  0. 

Any  covariant  with  n  odd,  m  =  2fi  >  0,  differs  from  one  of  rank 
>  m  by  a  linear  comhination  of  JQi'^i"  (/  =  1,  A,  A),  7v"*i", 
A7i:«Z"  and,  if  n>  I,  Qi'^-^L^'-W. 

28.  For  m  =  2n  >  0,  71  =  4^  >  0,  the  coefficients  of  ^2'"a*3"  in 

Ql'^<32^    K"^Q2,    Qi'-F-^    Qi-L\    K-^L-, 

are  respectively 

1,     «2,     63,     (33  +  1,     a.{h3  +1),     d  =  i83(/33  +  1),     aod. 

These  may  be  multiplied  by  any  invariant.     Xow 

1^3  +  1  +  Oo  +  63  =  aidi, 

A(/33  +  1)  +  (A  +  .1  +  1)63  +  h3a2  +  A  =  hzaicco, 

rf  +  .4  +  /33  +  «2(A  +  63)  =  hi{h3  +  «2)  =  /3, 

a2cZ  +  0263  =   «2&1&3  =   «2/3,       ^'W  =   ^61(63  +   1)   =   Afi. 

Hence  we  have  a  covariant  (76)  in  which  the  coefficient  of  ^i^Xz* 
is  any  linear  combination  of  the  functions   (71).     Hence  the 


94  THE  MADISON   COLLOQUIUM. 

covariant  differs  from  one  of  rank  >  m  by  a  linear  function  of  the 
covariants  (92),  the  'products  of  the  first  three  by  any  invariant 
except  1,  the  products  of  the  fourth  and  fifth  by  A  and  the  product  of 
the  sixth  by  A. 

29.  For  m  =  2/z  >  0,  n  =  ^v  +  2,  the  coefficients  of  ^2"*^3"  in 

(93)  MK^^-^q.^    K^L--,    GK^^-'Qi",    F'"-'Qi^,    L^Q^^ 
are  respectively 

G2,     02(&3+l),     azbsibi -j-  1) ,     bs,     63  +  aiaa  +  1. 

Linear  combinations  of  products  of  these  by  invariants  give* 

a2,     a^A,     a^J,     a^bz,    Aa^bz,     026163,     ^63,     (iioci,     A  +  63^10:2. 

Since  S  and  Si  are  unaltered  by  the  group  F  of  §  15,  they  are 
linear  combinations  of  the  functions  (71).  Deleting  the  above 
functions  a2,  a2A,  •  •  •  from  S,  we  have 

5  =  7  +  c/3  +  eAfi,    (3  =  61(63  +  0:2), 
where  c  and  e  are  constants,  and  I  is  an  invariant.     Set 

/l  =   BX2^  +  5i.T2"-^T3  + \-  Bn-lXoXz''-^  +  BnXz^, 

and  call  a  the  coefficient  of 

(94)  XiXiXz  ■  X2^''-^''-W~^  =  (xi^xsyxi^'-^xi 

in  xiXoXzcj).  The  coefficientf  of  (94)  in  R'  of  §  27  is  Bi  +  a. 
Hence 

<r'  -  a  =  Bu 

if  (50)  replaces  a  by  <t'.     Thus  J5i  must  be  of  the  form  (57). 

For  n  =  2,  JS2  is  derived  from  S  by  applying  (aia3)(6i63). 
Then  (672)  gives  Si.    Applying  (aia2)(6i62)  to  *Si,  we  get 

5i  =  /  +  0(6263  +  620;!  +  63ai)  +  eAib^bz  +  62  +  63). 


*  For  the  last  two,  use  the  first  two  of  the  four  equations  in  §  28. 

t  The  first  part  of  R'  is  free  of  xi,  the  second  of  Xz,  while  in  the  third  part 
^■?  has  the  factor  Xi^,  and  in/i'^i^*'  there  is  a  single  term  (94)  and  it  has  the 
coefficient  Bi. 


INVARIANTS   ANT)   NtTMBER   THEORY.  95 

Since  this  must  be  of  the  form  (57),  we  get  I  =  0,  c  =  e  =  0. 
A  covariant  with  m  =  2/2,  n  =  2,  differs  from  one  of  rank  >  mhy  a 
linear  function  of 

iMK'^-\    K^U,    AK'^U,    GK'^-K    IFq^\    UQ,'^,    ADQi'^ 

{i  =  1,  A,  J;  1=  1,  A,  A,  J). 

For  n>  2,  we  may  delete  A  from  the  part  /  of  »S  by  use  of 
EQi'^Qt"-'^,  where  E  is  given  by  (75).  Without  disturbing  S  we 
may  delete  02(63  +  1)  and  its  product  by  A  from  Si  by  use  of 
j^2^+ij^n-z^  since  the  term  of  ^i'^f^  with  the  coefficient  Si  is 
the  term  of  highest  degree  in  x^  in  ^a^^+H-Si-rs""^  +  •••)• 
Since  5  +  *Si  is  a  linear  combination  of  the  functions  (74), 

Si  =  S  -\r  tx{hz  -\-  aa)  +  haiai  +  Uhzaiao  +  ^463.4  +  t^an 
(95) 

+  ^A(63  +  1)  +  ^0(63  +  02)/. 

Apply  (aia2a3)(6i62&3)  to  Bi,  of  the  form  (57).     Hence 

(96)     Si  =  ppai  +  ^0262  +  pa2P  +  ra-z  +  sp,     p  =  61  +  03. 

Now  aib2  occurs  in  S  only  in  the  terms  J,  AJ  of  I  and  in  the  part 
of  (95)  after  S  only  in  the  last  term,  given  by  (72).  In  these  the 
factors  of  aib2  are  linearly  independent.  Hence  ^0  =  0, 
I  =  x(A  +  1).  The  coefficient  of  cti  in  ^i  must  vanish  for 
61  =  a3,  and  Si  itself  if  also  a2  =  0.     Hence 

c  =  t2  =  X,     h  =  ts  =  U  =  t,     U  =  X  +  t, 

Si  =  x{A  +  1  +  6163  +  hia2  +  aia2  +  Aa;2)  +  ^.461(63  +  1) 

+  t{hz  +02  +  hzaia2  +  ^3^4  +  63A  +  a^A). 

Call  €  the  coefficient  in  xiX2Xz4>  of 

a-ia;2a:3  •  .T2"^a-3^'^+"-3  =  {X2WYX1X2XZ''--. 

In  R'  of  §  27,  the  coefficient  of  this  product  is  e  +  5„_i.  Hence 
Bn-i  is  of  the  form  (57).  Interchanging  the  subscripts  1  and  2 
in  Bn-i,  we  get  »Si.  Thus  the  coefficient  of  a^  in  jSi  vanishes  for 
63  =  oi.     Hence  S  =  Si  =  0.     Any  covariant  with  n  >  2  differs 


9G  THE   MADISON   COLLOQUIUM. 

from  one  of  rank  >  m  by  a  linear  combination  of 

(i  =  1,  A,  J;  i  =  1,  A;  /  =  1,  A,  A,  J). 

CovAEiANTS  OF  Odd  Rank  ??Z  =  2/X  +  1  >  1,   §§  30-31 

30.  Replacing  xz  by  Xz  +  xi  in  the  co variant  (76),  we  get 

In  XiX2Xz(t>,  let  g  be  the  coefficient  of  {xiX^){x'^Xz)^~'^X2'^.  The 
coefficient  of  the  corresponding  term  of  R'  is  g'  ^  g  -\-  B,  where 
B  is  that  of  .^2"  in  /i.     Hence  B  is  of  the  form  (57). 

First,  let  n  be  odd.  Then  Si  =  Si-{-  S  under  (50),  so  that 
S  is  a  linear  combination  of  functions  (74)  with  a2(&3  +1)  and 
its  product  by  A  deleted  (§  23).  Thus  S  is  the  sum  of  the  terms 
(95)  after  the  first.  Applying  (aia2a3)  (616263)  to  B,  of  the  form 
(57),  we  see  that  *S  is  of  the  form  (96).     By  these  two  results, 

'S  =  ^(63  +  02  +  63aice2  +  63.4  4-  63A  +  a2A). 

If  I  is  the  coefficient  of  (x2X3-)^xz'^~^Xi  in  XiX2Xz(f>,  that  in  R'  is 
I'  =  I  -\-  nBn.  Hence,  for  n  odd,  Bn  is  of  the  form  (57).  Inter- 
changing the  subscripts  1,  2  in  Bn,  we  get  S.  Thus  the  coefficient 
of  ttz  in  »S  vanishes  for  63  =  ai,  so  that  i  =  0.  Any  covariant  with 
m  and  n  odd  differs  from  one  of  rank  >  m  by  a  linear  function  of 
K^^L""  and  Ai^"»X". 

31.  Finally,  let  m,  be  odd  and  n  even.  According  as  n  =  4i' 
or  4v  +  2,  K'^Q.J  or  K'^-^MQi"  is  of  the  form  (76)  with  02  as 
the  coefficient  of  ^2^xz'^.  Hence  we  may  delete  the  terms  aoli 
in  (85)  and  hence  the  terms  aili  in  5  of  §  23.  But  (§  30),  B  is 
of  the  form  (57).  Now  0361  occurs  in  J  and  A  J  of  I  and  in 
62 J  of  62/2,  having  in  these  linearly  independent  multipliers. 

Hence 

1=  x{A+l)  +  7/A,    L2  =  e-\-fA  +  gA. 

Since  the  coefficient  of  az  in  B  shall  vanish  for  63  =  ao,  and  B 
itself  if  also  Ui  =  0,  we  get  ki  =  x  =  y  =  kz,  ko  =  f  =  0  =  e. 


im^ARIANTS   AND    NUMBER  THEORY.  97 

Thus 

S  =  x(A  +  1  +  A  +  01^2  +  biba  +  bia2)  +  hiaihihz 

+  SrU  +  1  +  A  +  «ia2)&3  +  hAh,(h,  +  1). 
First,  let  n  =  'iv  -\-  2  and  write  2;u  +  1  for  m.     Then 

have  d  =  (Ssifii  +  1)  and  026?  as  the  coejSicients  of  ^2*^X3''.  As 
in  §  25,  the  coefficients  of  x,  k^,  g,  h  in  (97)  equal  respectively 

(?  +  aoCA  +  &3  +  1),     ttid-i-  a^hz,    Ad  +  a^d  +  02/,     Ad. 

The  terms  not  containing  d  are  combinations  of  the  above  a2li 
and  02(^3  +  1)  of  §  23.  Any  covariant  with  m  =  2ju  +  1  >  1, 
n  =  ^v  -\-  2,  differs  from  one  of  rank  >  m  by  a  linear  function  of 

{i  =  1,  A;  h  =  1,  A,  J;  7  =  1,  A,  A). 

Next,  let  71  =  4iV  >  0.  In  the  last  two  co variants  of  the 
theorem  below,  the  coefficients  of  ^•i^'^'^'^Xz^"  are  a^bzibi  +  1)  and 
5  =  bz^zi^i  +  1).  We  had  reached  covariants  in  which  the 
corresponding  coefficients  are  a^I  and  a^ibz  +1)7.  Thus  we 
obtain  the  coefficient  of  ^4  in  (97)  and  5  +  Aa2&3  +  ciibibz,  which 
equals  the  coefficient  of  g.  We  may  therefore  set  k^  =  g  =  0. 
Subtracting  covariants  of  the  fourth  and  fifth  types  in  the 
theorem,  we  may  take  as  Si  the  function  in  §  24,  without  dis- 
turbing S.  Applying  (aia2)(6i62)  to  S  and  ^Si,  we  get  5„  and 
Bn-i-  If  I  is  the  coefficient  of  XiX2^^^xz'^"*^"'~^  in  XiX^Xzff),  its 
coefficient  in  R'  of  §  30  is  Z'  =  Z  +  5„  +  5„_i.  Thus  Bn  +  Bn-i 
is  of  the  form  (57).  By  the  coefficient  of  0361,  if4=0.  Since  the 
coefficient  of  az  is  zero  for  63  =  a^,  we  get  x  =  k^  =  tz  =  0. 
Thus  S=  0.  Any  cotariant  ivith  m  =  2)u  +  1  >  1,  ?i  =  41/  >  0, 
differs  from  one  of  rank  >  m  by  a  linear  function  of 

K'"L'',  AX«i",  /Z'^Qo",  ^X"-3Ql'^+^  iL^-^K~'-+\  G^Kqi'-^qi'^-^ 

FGq^'-^qi'^     a  =  1,  .4,  A;  Z  =  1,  A,  J). 
8 


98  THE   MADISON   COLLOQUIUM. 

32.  We  have  now  completed  the  proof  of  the  theorem: 

As  a  fundamental  system  of  modular  covariants  of  the  ternary 
quadratic  form  F  ivith  integral  coefficients  modulo  2,  we  may  take 
F,  its  invariants  A,  A,  J,  its  linear  covariant  L,  its  "polar"  cubic 
covariant  K,  and  the  universal  covariants  Q\,  Q2,  L3. 

Incidentally,  we  have  obtained  a  complete  set  of  linearly 
independent  covariants  of  each  order  and  rank.  We  might  then 
find  a  complete  set  of  independent  syzygies.  Syzygies  whose 
members  are  covariants  of  low  rank  are  given  in  (78),  (88),  (91). 

33.  References  on  Modular  Geometry. — Other  aspects  of  the 
modular  geometry  of  quadratic  forms  modulo  2  and,  in  particular, 
applications  to  theta  functions  have  been  considered  by  Coble.  * 
For  a  treatment  of  non-homogeneous  quadratic  forms  in  x,  y 
modulo  p  {p  an  odd  prime),  analogous  to  that  of  conies  in 
elementary  analytic  geometry,  but  emplojdng  only  real  points  on 
the  modular  locus,  see  G.  Arnoux,  Essai  de  Geometric  analytique 
modulaire,  Paris,  1911.  The  earlier  paper  by  Veblen  and  Bussey 
was  cited  in  §  7.  The  paper  by  Mitchell  was  cited  in  §  3.  Appli- 
cations of  modular  geometries  have  been  made  by  Conwell.f 

The  problem  of  coloring  a  map  has  been  treated  from  the 
standpoint  of  modular  geometry  by  Veblen.  J 

*  Transactions  of  the  American  Mathematical  Society,  vol.  14  (1913),  pp. 
241-276. 

^Annals  of  Mathematics,  ser.  2,  vol.  11  (1910),  pp.  60-76. 
t  Annals  of  Mathematics,  ser.  2,  vol.  14  (1912),  pp.  86-94. 


LECTURE  V 

A  THEORY  OF  PLANE  CUBIC  CURVES  WITH  A  REAL  INFLEXION 
POINT  VALID  IN  ORDINARY  AND  IN  MODULAR  GEOMETRY 

1.  Normal  Forvi  of  Cubic. — Consider  a  ternary  cubic  form 
C{x,  y,  z)  with  coefficients  in  a  field  F  not  having  modulus  2  or  3. 
After  applying  a  linear  transformation  with  coefficients  in  F 
and  of  determinant  unity,  we  may  assume  that  (1,  0,  0)  is  an 
inflexion  point.  In  particular,  C  lacks  the  term  x^.  If  it  lacks 
also  x^y  and  x-z,  its  first  partial  derivatives  vanish  for  y  =  z=  0. 
But  we  shall  assume  that  the  discriminant  of  C  is  not  zero.  Hence 
the  coefficient  of  x-  may  be  taken  as  the  new  variable  y.  At  the 
inflexion  point  (1,  0,  0)  the  tangent  ?/  =  0  is  to  be  an  inflexion 
tangent,  i.  e.,  meet  the  cubic  in  a  single  point.  Hence  C  lacks 
the  term  xz-.     Thus 

C  =  x'-y  +  2x(ay'^  +  fiyz)  +  0(^,  z). 

Replacing  x  by  x  —  ay  —  I3z,  we  see  that  x'^y  is  now  the  only  term 
involving  x.  If  y  were  a  factor,  the  discriminant  would  be  zero. 
Hence  the  term  z^  occurs.  Adding  a  suitable  multiple  of  y  to  z, 
we  get 

(1)  C  =  x'y+gy'  +  kyh+dz'  (5  +  0). 

2.  The  Invariants  s  and  t. — The  Hessian  of  (1)  is 

H  =  -  Zbxh  -  hhf  +  95(72/-3  +  Wiyz^. 

The  sides  of  an  inflexion  triangle  form  a  degenerate  cubic  be- 
longing to  the  pencil  of  cubics  kC  +  H-     The  latter  has  the 
factor  z  only  when  k  =  h  =  0  and  the  factor  y  —  Iz  only  when' 
kl  =  35  (as  shown  by  the  terms  in  a*-),  where  k  is  a  root  of 

Ar^  +  185/?A;2  +  IQ^b-gk  -  275-/r  =  0. 

Before  considering  the  factors  involving  a-,  we  note  that  the 

99 


100  THE   SLADISOX   COLLOQUIUM. 

coeflBcients  of  this  quartic  equation  are  the  values  which  relative 
invariants  of  a  general  cubic  assume  for  the  case  of  our  cubic  (1). 
Indeed,  a  linear  transformation  of  determinant  unity  which 
replaces  C  by  a  cubic  C  must  replace  H  by  the  Hessian  H'  of  C", 
and  hence  replace  the  inflexion  triangle  of  C  given  by  a  root  k 
of  the  quartic  by  that  inflexion  triangle  of  C  which  is  given  by 
the  same  number  k.     We  denote  the  invariants  by* 

(2)  s=  -  38h,    t  =  -  1085-^. 
The  above  quartic  now  becomes 

(3)  k''  -  6sk-  -  tk  -  3s-  =  0. 
The  discriminant  A  of  C  is  such  that 

(4)  27A  =  f-  QisK 

There  are  four  distinct  roots  of  (3)  since  its  discriminant  is 
-  273A2. 

Our  earlier  results  are  that  kC  +  H  has  the  factor  z  only 
when  k  =  s  =  0  and  the  factor  y  —  3dk~h  if  ^  is  a  root  =f=  0  of 
(3) .     It  has  the  factor  x  —  ry  —  pz  if  and  only  if 

3p2  =  k,     9d'-kr~  =  s'-  +  tk/U,         kp'  -  UpT  =  s, 

Qdkpr  -  95 V2  -  sk  -  ^/4  =  0. 

These  conditions  are  satisfied  if  and  only  if  k  is  a  root  of  (3)  and 

p  =  ^.  =  0,     365-r-  =  -t  (A:  =  0), 

3p2  =  A-,     65A-r  =  p(il'2  -  3^)  (k  +  0). 

3.  The  Four  Inflexion  Triangles. — First,  let  s  =  0.  Then 
i  =}=  0  by  (4).  The  root  k  =  0  gives  the  inflexion  triangle  with 
the  sides 

(5)  z=  0,    x=  ±  ny  mSW  =  -  0- 


*  We  have  s  =  -  3*5',  t  =  -  3«r,  where  S  and  T,  given  in  Sahnon's  Higher 
Plane  Curves,  p.  189,  are  the  invariants  of  the  general  cubic  with  multinomial 
coeflBcients. 


INVARIANTS  AND  NUMBER  THEORY.  101 

Each  root  of  k^  =  t  gives  an  inflexion  triangle 

(6)  y  =  J'^'        ''=^''4^  +  rV    (3-366V  =  0. 
Next,  let  5  =#  0.     Each  root  of  (3)  gives  an  inflexion  triangle 

35  ^     [kf     ,   A;2  -  Ss    \ 

(7)  y  =  -j^z,        .^.=  ±^-^^,  +  -^^yj. 

4.  The  Parameter  8. — If  we  multiply  x,  y,  z  by  p,  p~^,  p,  we 
obtain  from  (1)  a  cubic  with  5  replaced  by  dp^.  If  F  is  the  field 
of  all  complex  numbers,  the  field  of  all  real  numbers,  or  the  finite 
field  of  the  residues  of  integers  modulo  3j  +  2,  a  prime,  every 
element  is  the  cube  of  an  element  of  the  field  [in  the  third  case, 
e  =  {e~^y],  so  that  the  parameter  d  may  be  taken  to  be  unity. 
Although  we  do  not  use  the  fact  below,  it  is  in  place  to  state  here 
that  for  all  further  fields  a  new  invariant  is  needed  to  distinguish 
the  classes  of  cubics  (1).  Indeed,  two  cubics  (1),  with  coef- 
ficients in  F  and  with  the  same  invariants  s  and  t  and  discrim- 
inants not  zero,  are  equivalent  under  a  linear  transformation 
with  coefiicients  in  F  and  having  determinant  unity  if  and  only 
if  the  ratio  of  their  5's  is  the  cube  of  an  element  of  F. 

Criteria  for  9,  3  or  1  Real  Inflexion  Points,  §§  5-9 

5.  Inflexion  Points  when  s  =  0. — Let  /c  be  a  fixed  root  of  k^  =  t. 
Let  Ti  and  T2  be  fixed  roots  of  the  equations  at  the  end  of  (5) 
and  (6).     Then 

(Ti/r2)2  =  -  3  =  (1  +  2co)^     co=  +  CO  +  1  =  0. 

Choose  w  so  that  Ti/r2  =  1  +  2a;.  Denote  the  lines  2=0, 
X  =  ny,  X  =  —  Tiy  in  (5)  by  Xi,  L2,  L3.  For  each  value  of 
i  =  0,  1,  2,  denote  the  three  lines  (6)  with  k  =  Koi'  by  Ln,  Liu 
Lzi,  that  with  the  lower  sign  being  Lzi.  Then  the  9  inflexion 
points  and  the  subscripts  of  the  4  inflexion  lines  through  each 
are  given  in  the  following  table: 


102 


THE   MADISON   COLLOQUIUM. 


(1,0,0)     i   (72,  1,0)      (-TO,  1,0) 


(8) 


10 

11 

12 


1 

20 
21 
22 


30 
31 
32 


(^^'^'35) 

2 

(     "^'^'35) 

3 

l2 

l2 

2,2-1 

2,2-2 

3,2-2 

3,  2  -  1 

In  the  last  two  columns,  i  has  the  values  0,  1,  2;  while  2  —  1  or 

2  —  2  is  to  be  replaced  bj^  the  number  0,  1,  2  to  which  it  is  con- 
gruent modulo  3. 

WTien  F  is  the  field  of  all  real  numbers,  k  may  be  taken  to  be 
real,  while  just  one  of  the  numbers  ti  and  T2  is  real.     Hence 

3  and  only  3  of  the  9  inflexion  points  are  real.  The  same  residt 
is  true  if  i^  is  the  field  of  the  p  residues  of  integers  modulo  p, 
where  p  is  a  prime  3 j  +  2  >  2.  For,  k  may  be  taken  to  be 
integral  (§4),  while  co  is  imaginary  and  hence  —  3  is  a  quadratic 
non-residue  of  j^-  If  —  Hs  a  quadratic  residue,  ti  is  real  and  T2 
imaginary.     If  —  i  is  a  non-residue,  the  reverse  is  true. 

Next,  let  2>  =  3j  +  1,  so  that  o)  is  real  and  hence  —  3  a  quad- 
ratic residue.  By  (5)  and  (6),  ri  and  T2  are  both  real  or  both 
imaginary  according  as  —  Hs  a  quadratic  residue  or  non-residue 
of  p.  Hence  all  9  inflexion  points  are  real  if  and  only  if  —  Hs 
both  a  square  and  a  cube  and  hence  a  6th  power  modulo  p.  If 
—  ^  is  a  square  but  not  a  cube,  only  the  first  3  inflexion  points 
are  real.  If  —  ^s  a  quadratic  non-residue,  (1,  0,  0)  is  the  only 
real  inflexion  point. 

A  cubic  with  integral  coefficients  taken  modulo  p,  a  prime  >  3, 
with  at  least  one  real  inflexion  point  and  loith  invariant  s  =  0 
and  invariant  i  =f=  0,  has  9  real  inflexion  points  if  p  =  3j  -\-  I 
and  —  t  is  a  sixth  power  modulo  p,  a  single  real  inflexion  point  if 
p  =  3j  -f  1  and  —  t  is  a  quadratic  non-residue  of  p,  and  exactly 
3  real  inflexion  points  in  all  of  the  remaining  cases. 

For  example,  if  p  =  7  and  5  =  0,  ^  4=  0,  there  are  9  real  in- 
flexion points  only  when  i  =  —  1.     Taking  5  =  3,  xi  =  —  2, 


INVARIANTS   AND   NUIMBER  THEORY.  103 

r2  =  +  1,  K^  -  1,  we  get  u=2.  Thus  x-y  -  7/  +  32^  =  0 
has  the  9  inflexion  points  (1,  0,  0),  (1,  1,  0),  (—  1,  1,  0), 
(-2,  1,  3  .  20,  (2,  1,  3  •  20      a  =  0,  1,  2). 

6.  Inflexion  Points  ichen  5  =t=  0,  A  =t=  0. — These  are  (1,  0,  0) 
and 


(9) 


/     g-F  35  \ 


where  A:  ranges  over  the  roots  of  the  quartic  (3) .  We  seek  the 
number  of  real  roots  k  for  which  V--^  is  real.  In  order  that 
the  left  member  of  (3)  shall  have  the  factors 

(10)  A--  +  wk  +  /,     F  -  ick  +  m, 
it  is  necessary  and  sufficient  that 

(11)  I  -\-  m  —  w^  =  —  Qs,     {I  —  m)w  =  t,     Im  =  —  35-. 
Let  ^  =1=  0  (for  /  =  0  see  §  9).     Then  w  +  0  and 

(12)  21  =  2f^  -  6*  +  tjiv,    2m  =  vf-  -  %s  -  t/iv. 
Inserting  these  values  into  (II3),  we  get 

(13)  w^  -  12sic^  +  48s-ir  -  f  =  0. 
Set  w~  =  y  -{-  4:S.     Then 

(14)  y^  =  f-  645^  =  27A. 

For  the  rest  of  this  section,  let  the  field  be  that  of  the  residues 
of  integers  modulo  p,  where  y  is  an  odd  prime  3J  +  2.  Since 
any  integer  e  has  a  unique  cube  root  e~^  modulo  p,  there  is  a 
single  real  root  2/  of  (14). 

First,  let  y  -\-  ^s  be  a  quadratic  residue  of  y.  Then  lo  is 
real  and  hence  also  /  and  m.  The  product  of  the  discriminants 
of  the  quadratic  functions  (10)  is  seen  by  (lli)  and  (II3)  to  equal 

(15)  (u'2  -  4/)(^/>2  _  4^„)  _  _  3(,j^,2  _  4^)2  =  _  3^2 

and  hence  is  a  quadratic  non-residue  of  p.  Thus  a  single  one  of 
the  quadratics  (10),  say  the  first,  has  a  discriminant  which  is  a 


104  THE   MADISON   COLLOQUirM. 

quadratic  residue  and  hence  has  real  roots.     By  (12i), 

4l(iv^  -  4Z)w2  =  _  2iv^  -  QicH  +  365Z(;4  -  if  +  48stw  -  lUshv\ 

Adding  the  vanishing  quantity  (13),  we  see  that 

(16)  4/(^2  -  4/)?(;2  =  _  3(m;3  _  g^^^,  _^  ^)2^ 

Since  w^  —  4^  is  a  quadratic  residue  and  —  3  is  a  non-residue 
of  p,  it  follows  that  Z  is  a  non-residue.  Hence  a  single  one  of 
the  roots  of  the  first  quadratic  (10),  and  hence  a  single  one  of 
the  roots  of  the  quartic  (3),  is  the  negative  of  a  quadratic  residue. 
Thus  just  two  of  the  inflexion  points  (9)  are  real. 

Next,  let  2/  +  4s  be  a  quadratic  non-residue  of  p.  Then  there 
is  no  factorization  of  the  quartic  (3)  into  real  quadratic  factors. 
Nor  is  there  a  real  linear  factor  k  —  r  and  a  real  irreducible 
cubic  factor.  For,  if  so,  the  roots  of  the  latter  are  of  the  form 
X,  X^,  X^"  (cf.  the  first  foot-note  p.  37).     Then 

(r-X)(r-XP)(r-XP'),  P=(X-X^)(X^-X''')(X^'-X)  =  P^    (modp) 

are  real,  so  that  the  discriminant  of  (3)  is  a  quadratic  residue. 
But  this  discriminant  was  seen  to  be  —  3(81A)-,  and  —  3  is  a 
non-residue.  Hence  (3)  is  irreducible  modulo  p.  Thus  (1,  0,  0) 
is  the  only  real  inflexion  point. 

For  p  =  Sj  -\-  2  >  2,  a  cubic  (1)  icith  stA  4=  0,  has  exactly 
three  real  inflexion  points  or  a  single  one  according  as  the  real 
number  3A'  +  4*  is  a  quadratic  residue  or  non-residue  of  p. 

7.  Cubic  with  stA  4=  0,  p  =  3j  +  1. — Now  —  3  is  a  quadratic 
residue  of  p  and  there  are  three  real  cube  roots  1,  co,  co^  of  unity 
modulo  p. 

In  this  section  we  shall  assume  that  A  is  a  cube  modulo  p. 
Then  there  are  three  real  roots  yi  of  (14).  At  least  one  of  the 
yi  +  4s  is  a  quadratic  residue  of  p  since 


n  (yi  +  4s)  =  2/1^  +  64s3  =  t'. 
If  2/1  +  4s  is  a  quadratic  residue,  while  2/2  +  4s  and  2/3  +  4s 


INVARIANTS   AND   NUMBER   THEORY.  105 

are  non-residues,  there  is  a  single  factorization  of  quartic  (3) 
into  real  quadratics  (10)  and  hence  certainly  not  four  real  roots. 
The  product  (15)  of  the  discriminants  of  the  real  quadratic 
factors  is  now  a  quadratic  residue  of  p.  If  each  were  a  residue, 
there  would  be  four  real  roots.  Hence  each  is  a  non-residue  and 
there  is  no  real  root.  There  is  a  single  real  inflexion  point  if 
2?  =  3j  +  1,  5/A  4=  0,  A  is  a  cube,  and  if  the  three  values  of  3A^  +  4* 
are  not  all  quadratic  residues  of  2^- 

Next,  let  each  yi  -f-  4s  be  a  quadratic  residue  of  p.  Then  there 
are  three  ways  of  factoring  quartic  (3)  into  real  quadratics  (10). 
But  a  root  common  to  two  distinct  real  quadratics  is  real.  Hence 
aU  four  roots  are  real.  The  discriminant  of  each  quadratic  (10) 
is  therefore  a  quadratic  residue  of  p.  Hence,  by  (16),  I  is  a 
quadratic  residue  of  p;  similarly  for  the  constant  term  of  each 
quadratic  factor.  Thus  the  negatives  of  the  four  roots  are  all 
quadratic  residues  or  all  non-residues. 

To  decide  between  these  alternatives,  we  need  the  actual  roots. 
In  Wi^  =  yi  +  45,  let  the  signs  of  the  Wj  be  chosen  so  that 

P  -  Wik  +  m;  =  0  (^  =  1,  2,  3) 

have  a  common  root.     As  in  (12), 

2m  i  =  IV  p  —  Qs  —  tjiVi. 

For  e  4=  1,  we  find  by  subtraction  and  cancellation  of  tvi  —  We 
that 

2k  =  Wi  -{-  10 e  +  t/{WilVc). 

Comparing  the  results  for  e  =  2  and  e  =  3,  we  get 
(17)  W1IV2WZ  =  t. 

Hence*  the  roots  of  (3)  are 

|(U'i  +  Wi  +  U's),       l{lOi  —  W2  —  W3), 
i{—  Wi-\-  W2  —  ICz),       i(—  Wi  —  ICo  +  W3). 

The  product  of  the  first  and  {i  +  l)th  roots  is  seen  to  equal  rrii 


In  particular,  we  have  deduced  Euler's  solution  by  the  method  of  Descartes. 


106  THE   MADISON   COLLOQUIUM. 

and  hence  is  a  quadratic  residue.  For  given  values  of  p,  s,  t, 
we  can  readily  find  by  a  table  of  indices  the  real  values  of  the  Wi 
and  thus  a  real  root  and  hence  decide  whether  or  not  it  (and 
hence  each  of  the  four  roots)  is  the  negative  of  a  quadratic 
residue. 

However,  changing  our  standpoint,  we  shall  make  an  explicit 
determination  of  all  sets  s,  t  for  which  the  quartic  (3)  has  four 
real  roots  each  the  negative  of  a  quadratic  residue  of  y. 

By  the  definition  of  the  u\^,  or  direct  from  (13), 

(19)  2tt'i2  =  125,     ^loi^wi^  =  48^-,     wxho^hoz^  =  t\ 

Let  w  be  a  fixed  integral  root  of  co^  +  co  +  1  —  0  (mod  p).     Then 

0  =  (12^)-  -  3(485^)  =  Swi^  -  2wiW 

=  {ic-^  +  coit?2"  +  orwz)  (wr  +  (x>ho%  +  wwz) . 
Interchanging  ^V2  and  Wz,  if  necessary,  we  have 

(20)  Wi^  +  wit'o^  +  u)hvz^  =  0         (mod  p). 

Conversely,  if  the  ivi^  are  any  quadratic  residues  satisfying 
(20)  and  if  we  define  s  and  t  by  (19i)  and  (17),  we  obtain  a  quartic 
(3)  with  the  four  real  roots  (18).  If  we  permute  wi,  w^,  lOz 
cyclically  we  obtain  solutions  of  (20)  leading  to  the  same  s  and 
t  and  to  the  same  four  roots  (18). 

Our  first  problem  is  therefore  to  find  all  sets  of  solutions  of 
(20).  To  this  end  it  is  necessary  to  treat  separately  the  cases 
—  1  a  quadratic  residue  and  —  1  a  non-residue;  viz.,  p  =  12^+  1 
and  p  =  12(7+  7  (since  already  p  =  3J+  1). 

First,  let  p  =  12q  -\-  1.  Then  —  1  =  i^  (mod  p),  where  i  is  an 
integer.     Set 

2p  =  «'i  —  iojicz,     2(r  =  Wi  +  iuwz. 

Then  (20)  becomes 

4pa  =  —  o)iV2^  =  {ioi^Wify 

so  that  per  must  be  a  quadratic  residue.     Hence  we  may  take 


INVAKIANTS   AND   NUMBER  THEORY.  107 

a  =  pV-,  where  p  and  /  are  integers  not  divisible  by  p.     Then 

(21)  Wi  =  p(l  +  V-),     U'.2  =  2ioipl,     Wz  =  2co2p(l  -  /2). 

We  must  exclude  the  values  of  I  which  lead  to  equal  values  of 
two  of  the  w'i",  and  hence  to  equal  t//s,  since  the  roots  of  (14)  are 
incongruent.  Now  if  any  two  of  the  w,^  in  (20)  are  congruent, 
all  three  are  congruent.     But  W\  =  w^^  implies 

l-\-l^=  ±2M,     (/=F^co)^=w^     /=±2a;  +  cco2       (e^  ^  1). 

The  values  Z-  =  0,  ±  1  make  one  of  the  lOi  ^  0.  Hence  we  must 
exclude  the  9  incongruent  integral  values  of  /: 

(22)  1  =  0,     rt  1,     d=  2',     CO-  ±  fco,     -  co2  ±  zco. 
Using  the  values  (21),  we  get 

(23)  125  =  p2{(l  -  co)(l  +  I')  -  Qd'P},     t  =  2pH{l'  -  1), 

('    7        \  2 
^  +  ]lT^)  • 

To  make  the  negative  of  the  last  a  square,  we  must  take 

(25)  p  =  -  2(1  +  2co2)r2  (r  ^  0). 
Now  s,  given  bj-  (23),  is  zero  only  when 

(26)  Z  =  CO  ±  ior,      —  co  =t  ?co-. 

The  desired  sets  s,  t  are  given  by  (23)  and  (25),  where  r  is  any 
integer  not  divisible  by  p,  tvhile  I  is  any  one  of  the  2^—13  positive 
integers  <  p  not  congruent  modulo  p  to  one  of  the  13  incongruent 
integers  (22),  (26).     The  minimum  p  is  37. 

Second,  let  p  =  12g  +  7.  Then  X-  =  —  1  (mod  p)  is  irre- 
ducible.    Its  roots  i  and  —  i  =  i^  are  Galois  imaginaries.     Set 

(27)  TT  =  2?  +  1,     a=p-l. 

There  exists  a  linear  function  R  of  i  with  integral  coefficients 
such  that  R""^  =  1,  while  no  lower  power  of  R  is  unity.  Any 
function  of  i  is  zero  or  a  power  of  i?  and  any  integer  is  a  power  of 


108  THE    MADISON   COLLOQUIUM. 

R",  a  primitive  root  of  p.     Hence  we  may  set 

where  0  ^  rj  <  a,  0  ^  e  <  ira.     Then  (20)  is  equivalent  to 

The  last  condition  is  equivalent  to 

(28)  e=2r]  +  <T/2+j(r  (0^i<7r). 

We  have 

W2  =  coR'\    2wi  =  R'  +  Rp',     2wz  =  -  m^R"  -  R^'), 

2o:''ZiVi  =  2R''^  +  (co2  -  i(^)R'  +  (co^  +  io,)R^\ 

(or  —  ico)  (or  -\-  io))  =  —  1, 
(29) 

(co-  -  io>y  =  -1,     d"  -  io:  =  Rf-'l^  (/  odd), 

2w22wi  =  2R''^  +  i?*+^"^2  -  Rv^-f'i^ 

=  R'''^J-U+i) I2\ (Rri-i+pU+i)  12  _|_  ]ipv-pj+(.f+i)/2y^ 

The  last  binomial  is  its  own  pih  power  and  hence  Is  real.  We 
desire  that  the  root  |2wi  shall  be  the  negative  of  a  quadratic 
residue  and  hence  a  non-residue.  Since  R"  is  a  primitive  root 
of  p,  the  condition  is  that  j  —  (/  +  l)/2  shall  be  odd: 

(30)  /  =  2/  -  1,    j-  1  =  odd. 

We  must  exclude  the  values  making  wi^  =  ivi^: 

0  =  2R''l\ivi  q=  W2)  =  R^^+'^+J'^  =f  2co7^"''+<^/2  _  i^2pw-^ 

the  second  term  having  been  simplified  by  use  of 

Jln,r/2   =    _    1,        RP-    =    R—, 

Completing  the  square  of  the  first  two  terms,  we  get 

(Rn+'^U+^)/2  ^  ^^JlPv-'Tjr-y.  =  (^2^  l)R'^P^-y. 

Now  co^  +  1  =  —  CO  =  (cioo^y,  where  c  =  1  or  —  1.     Hence 


INVARIANTS   AND   NUMBER  THEORY.  109 

But 

(co  +  ico2)(co  -  iV)  =  -  1,     CO  +  /co2  =  i?'"^ 
(31) 

CO  -  ICO^  =    -  i?-'"^/2  (^  O^J(J)^ 

Hence  we  must  exclude  the  four  cases  in  which 

(32)      rj^j+^{±v+l),    i  +  i(±t'+7r+l)         (mod  tt), 

these  four  values  being  incongruent. 

No  one  of  the  w's  in  (29)  is  zero,  since  e  is  odd  by  (28),  so  that 
e  #  0,  7r/2  (mod  tt).     By  (190  and  (17), 

48*  =  (1  -  o:){R-'  +  ii^pe)  4.  Gco^Es.", 
(33) 

4i  =  -  iR'^^R^'  -  R^P'). 

Finally,  we  must  here  exclude  the  cases  in  which  5=0. 
Combining  Swr  =  0  with  (20),  we  obtain  the  necessary  and 
sufficient  condition  Wi^  =  C0W3-  for  5  =  0.  But  Wi  =  d=  co^ws, 
in  connection  with  (29),  gives 

R'{1  ±  zco)  =  Rp'(-  1  ±  zco),     R'((^  ±  zco2)2  =  i^p^ 

Thus,  by  (31),  the  condition  is  that  e  =b  lu  =  pe  (mod  tto-)  or 
e  =  ±  V  (mod  tt).  Then,  by  (28),  7?  is  congruent  modulo  x 
to  one  of  the  values  (32)  decreased  by  ■7r/4.  Hence  the  desired 
sets  s,  t  are  given  by  (33),  subject  to  (28),  in  ivhich  the  8  incongruent 
r]'s  given  by  (32)  and  those  values  decreased  by  it  14:  are  excluded. 
In  particular,  p  >  7. 
For  p  =  19,  the  only  admissible  pairs  are 

,  5  =  2  '2'\    t  =  6(-  2)3^     {1=  0,1,  -..,8). 

For  any  I,  the  negatives  of  the  roots  of  quartic  (3)  are  the  products 
of  -  3  =  42,  4,  7  =  8^  -  8  =  72  by  (-  2)^  and  hence  are  quad- 
ratic residues  of  19  since  —  2=6^. 
For  p  =  31,  the  only  pairs  are 

5=32^,    i=5(_3)3i.    s=-3'-\     f=13(-3)3'     (;  =  0,  ••-,  15), 

the  negatives  of  the  roots  of  (3)  being  the  products  of  7,  —  11, 
—  12,  —  15  and  —  3,  5,  9,  —  11,  respectively,  by  (—  3)^  and 
hence  are  quadratic  residues  of  31. 


110  THE   MADISON   COLLOQUIUM. 

8.  Case  2^  =  3j  +  1,  stA  =1=  0,  A  not  a  Cube.— The  roots  of  (14) 
are  now  Galois  imaginaries  y,  y^,  y^'.    As  at  the  beginning  of  §  7, 

f-=  (y-{-  4s)  {y^  +  4s) (y^'  +  4^)  ^  (y  +  4s)'+p+^\ 

Raise  each  member  to  the  power  (p  —  l)/2.  We  see  that  y  +  4.9 
is  the  square  of  an  element,  say  w,  of  the  Galois  field  of  order  p^. 
The  first  root  (18)  is  ^{iv  +  w^  +  w^')  and  equals  its  own  pth 
power,  and  hence  is  real.  This  is  not  true  of  the  remaining  roots 
(18),  since  w'"  =1=  w,  or  since  a  real  quadratic  factor  would  imply 
that  w  is  real.     Hence  the  quartic  has  a  single  real  root. 

For  p  =  7,  the  only  cases  in  which  the  negative  of  the  single 
real  root  is  a  quadratic  residue  are  t  =  —  1  or  3,  5  =  —  1,  —  2,  3; 
t  =  2,  s  arbitrary  4=  0.     For  p  =  13,  the  only  cases  are 

±t=4,5,Q;    s  =  -  1,-3,4  (s^=-l); 

±t=l,5,Q;    5=-2, -5, -6  (*'=5); 

and  ±t=3,—s  equals  one  of  the  preceding  six  values  of  s. 

9.  Cubic  loith  t  =  Q,  s  ^  0. — In  this  case,  (3)  becomes 

(F  -  3sY  =  12s\ 
If  there  be  a  real  root  h,  3  is  a  quadratic  residue  of  p,  and 

/^2  =is^     /  =  3  d=  2  Vs. 

First,  let  p  =  3j  -\-  2,  so  that  —  3  is  a  quadratic  non-residue  of 
;;.  Then  —  1  must  be  a  non-residue  of  p  and  hence  p  =  12r  +  11 . 
The  product  of  the  two  Ts  is  —  3,  so  that  a  single  value  of  F  is 
a  quadratic  residue.  Since  the  two  real  k's  are  of  opposite  sign, 
there  is  a  single  real  root  h  whose  negative  is  a  quadratic  residue. 
For  /  =  0,  5  #  0,  and  j)  =  12r  +  5,  there  is  a  single  real  inflexion 
point;  for  p  =  12r  +11,  there  are  just  three  real  inflexion  points. 

Finally,  let  p  =  3j  +  1,  so  that  —  3  is  a  quadratic  residue  of  p. 
li  p  =  12r  +  7,  then  3  is  a  non-residue,  and  there  is  no  real  k 
and  hence  a  single  real  inflexion  point,  li  p  =  12r  +  1,  the 
four  roots  h  are  all  real  or  all  imaginary.  For  p  =  13,  P  =  —  2* 
or  —  55,  and  —  /c  is  a  quadratic  residue  if  and  only  if  k^  =  1, 
«3  =  8,  5  =  2,  5,  6.  For  p  =  37,  k^  =  -  4*  or  10^,  and  -  k 
is  a  residue  if  and  only  if  ***  =  1. 


TOPICS  IN  THE 

THEORY  OF  FUNCTIONS  OF 

SEVERAL  COMPLEX  VARIABLES 


BY 


WILLIAM   FOGG   OSGOOD 


CONTENTS 


LECTURE  I 
A  General  Survey  of  the  Field 

§  1.  Analytic  Functions  of  Several  Complex  Variables.  .  .  Ill 
§  2.     Jacobi's    Theorem    of    Inversion    and    the  Abelian 

Functions 113 

§  3.     Periodic  Functions 114 

§  4.     The  Theta  Theorem 118 

§  5.     Automorphic  Functions  of  Several  Variables 122 

§  6.     Continuation.     Hyperfuchsian     and     Hyperabelian 

Functions 125 

§  7.     Algebraic  Functions  of  Two  Variables 127 

§  8.     Analysis  situs 132 

LECTURE  II 

Some  General  Theorems 

§    L     Definitions  and  Elementary  Theorems 133 

§    2.     Line    and    Surface    Integrals,    Residues,    and    their 

Generalizations 135 

§    3.     The  Space  of  Analysis,  and  Other  Spaces 137 

§    4.     Rational  and  Algebraic  Functions 142 

§    5.     Sufficient    Conditions   that   a   Function   of   Several 

Complex  Variables  be  Analytic 142 

§    6.     Sufficient  Conditions  that  a  Function  be  Rational 

or  Algebraic 143 

§    7.     On  the  Associated  Radii  of  Convergence  of  a  Power 

Series 145 

§    8.     Hartogs's  Function  Rx 150 

§    9.     On    the    Analvtic    Continuation    of    a    Logarithmic 

Potential 153 

9  i 


11  CONTENTS. 

§  10.     The     Representation     of      Certain     Meromorphic 

Functions  as  Quotients 154 

§  11.     Integral  Functions  as  Products  of  Prime  Factors. .  .  .    156 

LECTURE   III 

Singular  Points  and  Analytic  Continuation 

§    1 .  Introduction 160 

§    2.  Non-Essential  Singularities 161 

§    3.  Essential  Singularities 163 

§    4.  Removable  Singularities 163 

§    5.  Analytic  Continuation  by  Means  of  Cauchy's  Integral 

Formula 165 

§    6.  Application  to  the  Distribution  of  Singularities 169 

§    7.  Generalizations  of  the  Theorem  of  §  5 171 

§    8.  Levi's  Memoir  of  1910 172 

§    9.  Continuation.     Lacunary  Spaces 175 

§  10.  Concerning  the  Boundary  of  the  Domain  of  Definition 

of/(.T,2/) 177 

§11.     A  Theorem  Relating  to  Characteristic  Surfaces 178 

LECTURE  IV 

Implicit  Functions 

§  1.     Weierstrass's  Theorem  of  Factorization 181 

§  2.     A    Tentative    Generalization    of    the    Theorem    of 

Factorization 184 

§  3.     Algebroid  Configurations 185 

§  4.     Continuation.     The  Branch  Points  of  the  Discrimi- 
nant     188 

§  5.     Single- Valued  Functions  on  an  Algebroid  Configura- 
tion     190 

§  6.     Solution  of  a  S^^stem  of  Analytic  Equations.     Weier- 
strass's Theorem 192 

§  7.     Continuation.     A  General  Theorem 193 

§  8.     The  Inverse  of  an  Analytic  Transformation 197 


CONTENTS.  Ill 

LECTURE  V 

The  Prime  Function  on  an  Algebraic  Configuration 

§    1.     The  Algebraic    Functions  of  Deficiency   1  and  the 

Doubly  Periodic  Functions.     GeneraHzations ....    199 

§    2.     The  Prime  Function  Defined  as  a  Limit 202 

§    3.     The  Existence  Theorems 204 

§    4.     Dependence  on  the  Parameter 207 

§    5.     The   Functions   in   the   Automorphic   Fundamental 

Domain 208 

§    6.     An  Auxiliary  Function 212 

§    7.     The  Prime  Function  ^(t,  t) 215 

§    8.     The     Determination     of     Q{t,  r)      by     Functional 

Equations 216 

§    9.     The  Abelian  Integrals  in  Terms  of  the  Prime  Function  218 

§  10.     The  Integral  of  the  Second  Kind  on  i^ 221 

§  11.     The  Integrands  of  the  Integrals  of  the  First  Kind.  .  .   223 

§  12.     The  Algebraic  Functions 225 

§  13.     Parametric  Representation  of  a  Homogeneous  Alge- 
braic Configuration 226 

§  14.     Linear  Differential  Equations  on  an  Algebraic  Con- 
figuration, and  the  Factor  (p'(t) 228 


TOPICS  IN  THE  THEOEY  OF  FUNCTIONS 
OF  SEVERAL  COMPLEX  VARIABLES 


BY 
WILLIAM  FOGG  OSGOOD 


LECTURE  I 

A  GENERAL  SURVEY  OF  THE  FIELD 

§  1.    Analytic  Functions  of  Seveeal  Complex  Variables 

In  the  decades  which  lay  between  Cauchy's  prime  and  the 
beginnings  of  the  modern  French  school,  the  theory  of  functions 
of  a  single  complex  variable  made  rapid  progress,  the  chief 
advances  taking  place  on  German  soil.  Simultaneously  with 
these  developments,  important  problems  in  the  theory  of  analytic 
functions  of  several  complex  variables  were  attacked  and  the 
theorems  connected  with  them  divined  with  an  insight  worthv 
of  the  genius  of  a  Riemann  and  a  Weierstrass. 

The  elementary  functions  of  several  real  variables  admit  exten- 
sion into  the  complex  domain  and  are  seen  to  be  developable 
there  by  Taylor's  theorem,  —  a  result  to  which  the  elementary 
theory  of  infinite  series  and  an  obvious  extension  of  Cauchy's 
integral  formula  alike  lead. 

It  was  natural,  then,  to  define  a  function  of  several  complex 
variables  generally  with  Weierstrass  as  one  which  can  be  de- 
veloped by  Taylor's  theorem  in  the  neighborhood  of  any  ordinary 
point  of  its  domain  of  definition;  or,  following  Cauchy,  as  one 
which  is  analytic  in  each  variable  separately  and  continuous  in 
all  taken  at  once.* 

*  Cauchy,  Turin  memoir,  1831,  =  Exercices  d'analyse,  2  (1841),  p.  55;  Jor- 
dan, Cours  d'analyse,  1,  2d  ed.,  1893,  §206.  The  condition  of  continuity  is 
introduced  to  simplify  the  proofs.     It  is  a  consequence  of  the  former  condition ; 

111 


112  THE   MADISON   COLLOQUIUM. 

The  Factorial  Function  and  Analytic  Continuatio?i.  One  of 
the  problems  with  which  mathematicians  had  occupied  them- 
selves without  obtaining  satisfactory  results  was  that  of  extending 
the  definition  of  the  function  n !  to  a  continuous  range  of  values 
for  the  argument.  This  question  Weierstrass*  took  up,  exam- 
ining the  work  of  his  predecessors  and  showing  that  a  satisfactory 
solution  could  be  reached  on  the  basis  of  the  principle  of  analytic 
continuation,  the  functions  considered  being  dependent  on 
several  variables.  Thus  these  functions  contributed  at  that 
early  time  to  the  recognition  of  the  importance  of  the  conception 
of  the  monogenic  analytic  configuration. 

Existence  Theorems.  Cauchy  had  established  the  first  existence 
theorems  for  ordinary  differential  equations  and  implicit  func- 
tions.! In  his  further  study  of  these  problems  he  developed  the 
method  of  power  series  and  series  majorantes.X 

The  extension  to  the  case  of  partial  difterential  equations  was 
direct,  and  the  results  thus  obtained  were  of  importance.  For, 
while  much  of  the  theory  of  these  equations  appeared  plausible 
from  geometric  considerations  of  a  somewhat  crude  sort  or  from 
analogy  with  special  examples  yielding  an  explicit  solution,  a 
secure  foundation  had  hitherto  been  lacking. 

Weierstrass' s  Theorem  of  Factorization.  If  a  mathematical 
theory  is  to  gain  its  independence  and  take  its  place  among  the 
powers,  it  must  recognize  its  own  peculiar  problems  and  obtain 
methods  for  dealing  with  them.  One  of  the  earliest  distinctive 
theorems  which  became  known  in  the  theory  of  functions  of 
several  complex  variables  is  the  theorem  of  factorization,  due  to 
Weierstrass.  § 

cf .  below,  Lecture  II,  §  5.  Such  citations  will  be  made  in  the  following  pages  as 
H,  §  5. 

In  order  not  to  interrupt  the  course  of  the  general  account  with  which  we 
are  now  engaged,  the  consideration  of  a  number  of  detailed  consequences 
which  follow  from  the  definition  will  be  postponed  to  a  later  paragraph;  cf.  II, 
§§1,2. 

*  Journ.fiir  Math.,  51  (1856),  p.  1;  Werkc,  1,  p.  153. 

t  Cf.  Enzyklopiidie  der  math.  Wiss.,  II  B  1,  p.  103,  and  ibid.  II  A  4a,  p.  201. 

J  Turin  memoir,  1831;  Excrcices  d'analyse,  1  (1840),  p.  327. 

§  Cf.  IV,  §  1.     The  theorem  dates  from  1860. 


FUNCTIONS    OF   SEVERAL   COMPLEX   VARLiBLES.  113 

By  the  aid  of  this  theorem  he  proved  the  extension  of  Riemann's 
theorem  relating  to  removable  singularities,*  at  least  for  the 
case  that  the  given  function  can  be  expressed,  in  that  part  of  the 
neighborhood  of  the  given  point  where  it  is  defined,  as  the 
quotient  of  two  functions  each  analytic  at  the  point. 

It  would  be  of  interest  to  know  whether  Weierstrass  ever 
considered  the  theorem  in  its  general  form.  I  recall  no  passage 
in  his  writings  which  contains  such  a  reference.  Is  it  possible 
that  the  restricted  form  just  mentioned  was  sufficient  for  all  the 
applications  of  this  important  theorem  which  he  met? 

§  2.    Jacobi's  Theorem  of  Inversion  and  the  Abelian 

Functions 

Toward  the  close  of  the  eighteenth  century  the  way  was 
paved,  through  Legendre's  researches  in  the  theory  of  the  elliptic 
integrals,  for  some  of  the  most  important  advances  which  have 
been  made  in  analysis  since  the  invention  of  the  calculus, — 
those  which  cluster  about  the  elliptic  functions  and  their  general- 
izations, the  Abelian  and  the  automorphic  functions.  Jacobi, 
following  a  line  of  thought  which  Abel  had  initiated,  was  led  to 
formulate  the  problem  of  inversion  which  bears  his  name.f 

The  first  solutions  of  this  problem  which  appeared,  restricted 
to  the  case  p  =  2, — those  of  Gopel  (1847)  and  Rosenhain 
(1846-51), — were  based  on  the  theta  functions  of  two  arguments.^ 
Weierstrass§  and  Riemann||  arrived  independently  at  solutions 
in  the  general  case  of  the  Abelian  integrals  corresponding  to  an 

*  Cf.  Ill,  §  4. 

t  Jacobi,  Considerationes  generales  de  transcendentibus  Abelianis,  1832; 
Ges.  Werke,  2,  p.  o.  For  a  statement  of  the  general  problem  cf.  Neumann, 
Abelsche  Integrale,  2d  ed.,  1884,  Chs.  14,  15;  Appell  et  Goursat,  Fonctions 
algebriques,  Ch.  10.  For  an  account  of  the  history  of  this  problem  cf.  Krazer's 
Festrede:  Zur  Geschichte  des  Umkehrproblems  der  Integrale,  Karlsruhe,  1908. 

t  Jacobi  and  Gopel  independently  extended  the  elliptic  thetas  to  the  thetas 
of  several  arguments;  cf.  Krazer,  1.  c,  pp.  17,  18. 

§  Beitrag  zur  Theorie  der  Abelschen  Integrale,  Braunsberg,  1849,  =  Werke, 
1,  p.  Ill;  Journ.fiir  Math.,  47  (1854)  p.  289  =  Werke,  1,  p.  133;  ibid.,  52  (1856), 
p.  285=  Werke,  1,  p.  297.     Also  Werke,  4. 

II  Journ.fiir  Math.,  54  (1857),  pp.  101/155  =  Werke,  1  ed.,  p.  81;  2d  ed.,  p.  88. 


114  THE   M.VDISON   COLLOQUIUM. 

arbitrary  algebraic  configuration.  In  these  investigations  both 
mathematicians  were  led  to  the  study  of  the  theta  functions  of 
p  arguments,  —  in  fact,  Weierstrass,  to  whom  the  generalized 
thetas  were  at  that  time  unknown,  thus  came  to  discover  the 
form  of  these  functions.* 

The  Abelian  functions  themselves  are  not  single-valued.  They 
are  the  roots  of  algebraic  equations  of  degree  p,  whose  coefficients 
are  single-valued  functions  having  only  non-essential  singularities 
in  the  finite  region  of  the  space  of  their  p  complex  arguments  and 
admitting  2p  independent  periods;  cf.  §  3. 

Here,  then,  is  a  general  class  of  functions  of  several  variables, 
to  which  Jacobi's  problem  of  inversion  has  directly  led, — the 
class  which  corresponds  to  the  doubl}^  periodic  functions  of  a 
single  variable. 

§  3.    Periodic  Functions 

To  state  more  precisely  what  is  meant  by  periodicity,  it  is 
this.  The  function  f(zi,  •  •  • ,  Zn)  is  said  to  admit  the  periodf 
(P)  =  (Pi,  •••,Pn)if 

/(Zl  +  Pi,  22  +  P2,    •••,Zn+  Pn)   =  /(Zl,    '  '  *,  Zn), 

where  Pi,  •  •  • ,  Pn  are  constants. 

AYe  shall  restrict  ourselves  here,  unless  the  contrary  is  explicitly 
stated,  to  functions  which  are  single-valued  and  have  no  other 
than  non-essential  singularities  (III,  §  2)  in  the  finite  region  of 
space. 

If  (P)  and  (Q)  are  two  periods,  then  (P)  +  (Q)  =  (Pi  +  Qu 
•••,  Pn-\-Qn)  is  evidently  also  a  period.  Moreover,  (— P) 
=  (—Pi,  •  •  •,  —Pn)  is  a  period. 

A  function  /(zi,  •  •  •,  z„)  is  said  to  be  /j-fold  periodic  if  there 
exist  k  periods  (P'),  iP"),  •  •  •  (P^^'O,  and  no  fewer,  in  terms  of 

*  For  their  definition  cf.  §  3. 

t  Weierstrass  uses  the  term  system  of  periods  (Periodensystem),  i.  e., 
simultatieous  system  of  periods,  to  denote  tliis  complex,  which  may  be  thought 
of  as  a  vector  in  space  of  2p  dimensions.  The  briefer  term  period  would  seem 
to  suffice. 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARL-VBLES.  llo 

which  every  period  (P)  can  be  expressed  linearly  with  integral 
coeflBcients : 

(P)  =  m'(P')  +  m"(P")  + h  in^'^P^''^). 

Such  a  set  of  periods  is  called  a  primitive  scheme,  or  set,  of  periods.* 
A  periodic  function  which  is  a  constant  or  which  depends  on 
fewer  than  ??  arguments  will  evidently  not  come  under  this 
definition.  This  will  also  be  the  case  if,  on  making  a  suitable 
non-singular  linear  transformation  of  the  arguments,  f(zi,  •  •  • ,  2„) 
goes  over  into  a  function  of  fewer  than  n  arguments.  All  other 
periodic  functions  do  come  under  this  definition,  the  functions 
excluded  being  precisely  those  which  admit  infinitely  small 
periods. 

It  is  a  theorem  due  to  Riemannf  that  a  A'-fold  periodic  function 
of  ^^-independent  variables  cannot  exist!  when  k  >  229.  On  the 
other  hand,  the  Abelian  functions  have  led  to  2p-fold  periodic 
functions  of  j^  complex  arguments,  and  such  functions  can  also  be 
formed  by  means  of  quotients  of  theta  functions  of  p  arguments. 
Theta  Functions  idth  Several  Arguments. — The  fundamental 
theta  function  of  a  single  argument!  can  be  defined  by  a  series 
as  follows: 

t?(w)  =  d{u,  a)  =  C  T,  e°"'^2n«^  C  +  0, 


where 

. 

a  =  r  -\-  si 

and 

r  =  ^(a)  <  0 

*  I  avoid  the  term  primitive  system  of  periods  because  of  the  confusion  which 
would  thus  be  introduced,  due  to  the  other  sense,  above  mentioned,  in  which 
the  words  system  of  'periods  are  used. 

\Journ.Sur  Math.,  71  (1859),  p.  197=Werke,  1  ed.,  p.  276;  2d  ed.,  p.  294. 
Cf.  also  Weierstrass,  Berliner  Monatsber.,  187G,  p.  680  =  Werke,  2,  p.  55. 

t  The  maximum  number  of  periods  which  an  integral  function  can  have  is  p. 
Hermite,  in  Lacroix's  Calcul  differentiel  et  calcul  integral,  vol.  2,  6th  ed.,  1862, 
p.  390. 

§  This  function  appears  in  Fourier's  Theorie  analytique  de  la  chaleur,  1822, 
p.  333.  It  is  usually  thought  of  as  due  to  Jacobi,  who  was  the  first  to  recognize 
its  importance  in  the  theory  of  the  elliptic  functions;  Fundamenta  nova,  1829, 
=Werke,  1,  p.  228. 


116  THE   MADISON   COLLOQUIUM. 

It  has  the  properties: 

^{u  +  Tvi)  =  ^{u), 
i?(w+  a)  =  e-2"-«t?(w); 

and  it  has,  moreover,  a  single  root  of  the  first  order  in  the  parallel- 
ogram F,  two  sides  of  which  are  the  vectors  (0,  wi)  and 
(0,  a). 

By  means  of  this  function,  doubly  periodic  functions  can  be 
formed  as  follows.  Let  cei,  •  •  • ,  a„,  /3i,  •  •  -,  fin  be  any  2n  points 
so  chosen  that 

n  n 

H(Xk  =    S  fik, 

and  that,  furthermore,  the  points  of  the  parallelogram  F  that 
are  congruent  to  them  are  distinct.     Then  the  quotient 

i}{u  -\-  ai)    •  •  •    t?(M  +  OCn) 
HU  +  iSl)    •  •  •    HU  +  fin) 

will  evidently  represent  a  doubly  periodic  function  with  the 

periods  iri  and  a. 

The  fundamental  theta  function  of  p  arguments  is  given  by 

the  following  series: 

p 

r+2  S  n*MA 

^(ui,  '•■,up)  =  CZe     *=*      ,  C4=0, 


where 


p 


r  =  r(wi,  •  •  -,  np)  =   J2  akinkUi,         aki  =  aikr 

k,  /=1 
(Ikl  =    Tkl  +  iSkl, 

and  the  real  part  of  T{xi,  •  •  •,  Xp),  where  xi,  -  ■  •,  Xp  denote  real 

variables,  namely 

p 

Z)    TklXkXi, 
k,l=\ 

is  a  definite  negative  quadratic  form. 

The  function  has  the  following  properties,  readily  deducible 


FUNCTIONS   OF   SEVERAL   COMPLEX   VAmABLES.  117 

from  the  series,*  which  we  write  at  length  for  the  typical  case 
p  =  3. 

«?(?/!  +  iri,  u.>,  Uz)  =  §{11.1,  112,  W3), 

t?(wi,  «2  +  tt/,  Uz)  =  d(ui,  7/2,  Uz), 
■&{ui,  ?/2,  iiz  +  Tri)  =  ??(2/i,  U2,  Uz), 

^{Ul  +  «ii,  «2  +  0-21,   Us  +  fi3l)    =    e~^"'~''"??(«l,   ?/2,   ?<3), 
t?(Wi  +  «12,  ?<2  +  «22,   W3  +  a32)    =   e~^"-~''*'?9(?/l,  Wo,  ws), 

t?(wi  +  ai3,  U2  +  ^23,  W3  +  a33)  =  e~^""'~''''^(ui,  U-i,  Uz). 

The  vectors  in  2p-dimensional  space  corresponding  to  the  2p 
columns  in  the  array 


ri    0      0 
(1)  0      7r^    0 

0      0      iri 


Oil     an     an 

Cf21       O22       Q23 
«31       O32       «33 


form  the  edges  of  a  true  prismatoid,  F,  and  a  periodic  function 
corresponding  to  F  can  be  formed  as  follows.  Let  2pn  =  Gti 
complex  numbers  an,  fiki,  k  =  I,  •  •  -,  p  =  S,  I  =  I,  •  •  • ,  n,  he 
so   chosen   that 


«11  +  • 

■    -\-  OCln   =   jSll  +    • 

•  •  +  ^m, 

CX21  +  • 

•    +  (Xin   =    ^21  +    • 

■  ■    +  ^2n, 

«31  +    • 

•  •    +  azn   =   /331  +    • 

■  ■    +  ^Zn, 

but  that  these  numbers  are  otherwise  non-specialized.     Then  the 
quotient 

t^(Ui-\-aii,    U2-\-a21,    ^3  +  0:31)    •  '  •    ^(Ul+ain,    U2  +  a2n,    M3+Q:3n) 
«?(Wl+|8ll,    U2+^2l,    Uz+M    •  •  •    ^{Ul+^ln,    W2+j82n,    Uz+^Zn) 

will  represent  a  function  admitting  as  a  primitive  scheme  of 
periods  the  above  scheme  (1).     It  is  sufficient  to  take  n  =  2. 

As  regards  the  proof  of  this  theorem,  it  is  clear  that  the  above 
quotient  admits  each  period  of  the  scheme  (1);  but  it  is  not 

*  Cf.  Ivrazer,  Lehrbuch  der  Thetafunktionen,  Chap.  1. 


lis  THE   MADISON   COLLOQUIUM. 

clear  that  the  a's  and  /3's  can  be  so  chosen  that  this  scheme  is 
primitive  for  the  function.     This  is,  however,  the  case.* 

A  second  mode  of  obtaining  2p-fold  periodic  functions  belong- 
ing to  the  scheme  (1)  is  as  follows.     The  functions 

52  log  ^ 


dzk  dzi 


k,   I    =    1,     '--yih 


obviously  admit  the  periods  of  (1),  and  it  is  readily  shown  that 
they  admit  only  such  periods  as  are  expressible  linearly  with 
integral  coefficients  in  terms  of  these.f  And  now  it  can  be 
proven  that  a  linear  combination  of  the  above  functions  can  be 
so  chosen  as  to  yield  a  function  belonging  to  the  scheme  (1). 
This  statement  is  made  by  Wirtinger,  1.  c,  but  the  proof  is  far 
from  obvious. t 

The  number  of  essential  constants  on  which  an  algebraic  con- 
figuration of  deficiency  p  >  1  depends  is  32:*  —  3, — the  so-called 
moduli.  For  p  —  2  and  2^  =  3  this  number  is  the  same  as  the 
number  of  complex  constants  in  the  theta  function,  namely 
^p{p  +  1).  But  for  2?  >  3  the  latter  number  is  larger,  and  hence 
the  Abelian  functions  of  p  arguments,  —  or  rather  the  symmetric 
functions  of  their  multiple  determinations,  —  are  not  the  most 
general  2p-fold  periodic  functions. 

§  4.    The  Theta  Tiieokem. 

Can  all  22J-fold  periodic  functions  with  only  non-essential  singu- 
larities in  the  finite  region  be  expressed  in  terms  of  theta  functions 
of  p  arguments?  The  answer  to  this  question  is  affirmative, 
and  is  the  noted  theta  theorem  due  to  Riemann  and  Weierstrass. 

At  first  sight  a  mere  count  of  constants  appears  to  discredit 
the  theorem.  For  the  general  theta  function  of  2>  arguments 
depends  on  but  ^pip  +  1)  complex,  or  x^ip  +  1)  real  constants, 
namely,  the  Uki  subject  to  the  equations  a^-^  =  aik,  while  the 
region  of  223-dimensional  space  which  is  the  analogue  of  the 

*Cf.  a  forthcoming  paper  by  the  author.     (Note  of  December  29,  1913.) 

t  WirtinfTor,  Monalshefte  f.  Math.  u.  Phys.,  G  (1895),  p.  96,  §  IG. 

t  Cf.  a  forthcoming  paper  by  the  author.     (Note  of  January  IS,  1914.) 


FUNCTIONS   OF   SEVERAL   COMPLEX  VARIABLES. 


119 


parallelogram  of  periods  for  ^J  =  1  and  which  forms  a  funda- 
mental region  for  the  function,  —  the  prismatoid,  /',  —  depends, 
after  reduction  to  normal  form,  as  we  shall  presently  see,  on  p^ 
complex,  or  2jr  real  constants. 

With  reference  to  this  normal  form,  let  21,  •  •  •  ,Zp  be  the  original 
arguments  and  let  the  original  2p  periods,  which  are  linearly  inde- 
pendent, be  WTitten  in  the  columns  of  the  following  array: 


(2) 


Then  at  least  one  of  the  p-rowed  determinants  taken  from  the 
matrix  of  the  2p2  ^^'g  corresponding  to  the  scheme  (2)  will  be  dif- 
ferent from  0.*  Let  this  be  the  determinant  ±  Scon  •  •  •  copp.  If 
now  we  set 

TviZk  =  OikXllX  +    •  •  •   +  CO^.p?/p, 


Zi 

cou      • 

•  •       COip 

£0u         • 

•  •       Wl'p 

22 

W21     • 

•  •        C02p 

W21         • 

•  •        W2p 

2p 

Wpi       • 

••        COpp 

COpl       • 

•                  • 

•        03pp 

the  scheme  of  periods  for  the  transformed  function  /(zi, 
=  F{u\,  • ' ' ,  Up)  will  be  as  follows: 


Zp) 


Ul 

Wl, 

0, 

' ' ' ) 

0 

an. 

•••)       Clip 

H2 

0, 

iri, 

' ' ' ) 

0 

021, 

'      • ,        (l2p 

Up 

0, 

0, 

... 
> 

7^^ 

dpi, 

'  '  '  }       0,pp 

Thus  we  have  in  the  aus  p^  complex,  or  2/?^  real  constants. 
If  these  be  given  non-specialized  values,  we  are  led  to  a  true 
2/>dimensional  prismatoid. 

To  any  n-dimensional  prismatoid  F  correspond  real  analytic 
functions  of  n  real  variables  with  n  periods,  for  which  F  is  a 
fundamental  domain.     If,  then,  in  the  case  before  us,  the  most 

*This  theorem  is  contained  in  a  paper  by  Kronecker,  Berliner  Sitzimgsber., 
29,  (1884),  p.  1071.  Its  proof  follows,  however,  readily  on  developing  system- 
atically the  elements  of  the  periodic  functions  of  several  complex  variables 
from  a  geometric  standpoint. 


120  THE   MADISON   COLLOQUIUM. 

general  22>fold  periodic  analytic  functions  of  y  complex  variables 
are  to  be  represented  by  means  of  quotients  of  thetas  with  p 
arguments,  this  means  that  the  prismatoid  is  here  subject  to 
essential  restrictions,  since  ^(p  +  1)  <  2;;-. 

That  this  is,  in  fact,  the  case  was  discovered  independently 
b}'  Riemann  and  Weierstrass,  and  thus  the  first  step  was  taken 
toward  the  establishment  of  the  theta  theorem. 

Riemann  never  published  a  proof  of  the  theorem.  He  com- 
municated his  results  to  Hermite*  in  18G0.  Weierstrass's  proof 
was  not  given  in  detail  till  the  appearance  of  his  collected  works,  f 
though  he  had  published  a  number  of  notes  bearing  on  a  proof, 
and  had  stated  the  theorem  in  a  letter  to  Borchardt.| 

In  the  early  eighties  Poincare  and  Picard§  constructed  proofs 
of  the  theorem,  which,  it  turned  out,  were  essentially  the  same 
as  Weierstrass's.  Appell||  gave  a  proof  in  1891  along  different 
lines.  Then  came  a  proof  by  Wirtinger,^  which  has  much  in 
common  with  Weierstrass's  proof,  at  that  time  unpublished. 

Shortly  after,  Poincare**  gave  a  new  proof,  in  which  the  method 
is  that  of  potential  functions  in  hyperspace.  Kroneckerff  had 
already  surmized  that  this  method  would  lead  to  fruitful  results 
in  the  theory  of  functions  of  several  complex  variables. |t  Poin- 
care had  used  this  method  in  an  earlier  paper,  in  proving  the 
theorem  that  a  function  of  two  complex  variables  which  has  no 
other  than  non-essential  singularities  in  finite  space,  can  be 
expressed  as  the  quotient  of  two  integral  functions,  and  that  this 
quotient,  moreover,  at  any  point  at  which  both  numerator  and 

*  Cf.  Lacroix,  Calcul  diffcrentiel  et  calcul  integral,  vol.  2,  6th  ed.,  1862, 
J).  390. 

t  Werke,  3,  1903,  p.  53. 

tJourn.fiir  Math.,  89  (1880),  p.  8  =  Werke  2,  p.  133.  Berliner  Monats- 
berichte,  1869,  p.  855=  Werke  1,  p.  46. 

§  C.  R.,  97  (1883),  p.  1284.     Poincar6,  Acta,  22  (1898),  p.  90. 

II  C.  R.,  110  (1890),  pp.  32,  181;  Journ.  de  Math.  (4),  7  (1891),  p.  157. 

t  Monatsheflef.  Math.  u.  Phys.,  6  (1895),  p.  69.     Cf.  also  ibid.,  7  (1896),  p.  1. 

**  Acta,  22  (1898),  p.  89.     Cf.  also  ibid.,  26  (1902),  p.  43. 

tt  Berliner  Monal.sberichte,  1869,  pp.  159,  688  =  Werke,  1,  p.  198. 

Xt  Cf.  two  mcmoins  by  Baker,  Transactions  Cambridge  Phil.  Soc,  18  (1900), 
p.  408,  and  Proceedings  London  Math.  Soc.    2),  1  (1904),  p.  14. 


FUNCTIONS    OF   SEVERAL   COMPLEX    VARL\BLES.  121 

denominator  vanish,  will  be  in  reduced  form.*  It  is  this  theorem, 
too,  on  which  Appell's  proof  cited  above  rests. 

All  of  these  proofs  involve  a  considerable  amount  of  analytical 
developments.  Weierstrass  was  led,  in  the  course  of  his  analysis, 
—  and  it  may  be  remarked  in  passing  that  he  edited  his  proof 
with  minute  care, — to  emphasize  the  importance  of  an  accurate 
definition  of  the  monogenic  analytic  configuration  of  the  mth 
grade  (Stufe)  in  the  domain  of  n  complex  variables.  He  points 
out  that  it  will  not  do  to  start  with  the  points  for  which  certain 
of  the  coordinates  chosen  as  dependent  variables  are  analytic  in 
the  remaining  coordinates  considered  as  independent  variables, 
and  then  adjoin  all  limiting  points  to  the  set  thus  obtained. 
For,  in  the  case  of  two  variables,  he  says,  it  may  happen  that 
one  would  thus  obtain  all  the  points  of  space,  f 

Furthermore,  in  the  proof  as  Weierstrass  originally  conceived 
it,  —  the  final  proof  which  appeared  in  his  collected  works  is 
modified  in  essential  respects,  —  two  general  theorems  relating 
to  periodic  functions  play  an  essential  role.     They  are  these,  i 

I.  Any  2p-fold  periodic  function  of  i)  variables  is  an  algebraic 
function  of  ;;  independent  2^>fold  periodic  functions  belonging 
to  the  same  prismatoid.     Or,  otherwise  expressed: 

Betw^een  any  p  +  1  2p-fold  periodic  functions  of  p  variables 
there  exists  an  algebraic  relation. 

II.  Any  2p4o\d  periodic  function  of  p  variables  is  expressible 
rationally  in  terms  oi  p  -\-  I  suitably  chosen  22>fold  periodic 
functions  belonging  to  the  same  prismatoid. 

These  theorems  have  been  generalized  by  Picard  and  Wirtinger 
for  automorphic  functions  of  several  variables;  cf.  §§5,  G. 

Poincare's  potential  functions  undoubtedly  form  a  powerful 
instrument  of  analysis  in  dealing  with  the  singularities  of  func- 

*Cf.  IV,  §1. 

t  Werke,  3,  p.  96. 

X  Berliner  Monatsberichle,  1869,  p.  855=  Werke,  2,  p.  46.  These  theorems 
have  been  treated  by  Poincare,  C.  R.,  124  (1897),  p.  1407;  Wirtinger,  Sitzungs- 
ber.  der  Wiener  Akad.,  108  (1899),  p.  1239;  and  Bkimenthal,  Math.  Ann.,  58 
(1904),  p.  497;  cf.  also  Math.  Ann.,  56  (1903),  pp.  510,  512. 


122  THE   MADISON   COLLOQUIUM. 

tions  of  several  complex  variables.  He  carries  his  proof  through 
only  to  the  point  of  showing  that  the  given  function  can  be  \VTitten 
as  the  quotient  of  two  Jacobian  functions.  The  latter  functions 
are  defined  as  follows. 

Jacobian  Functions.  Let  co^^,  a=  1,  •••,  j^l  0  —  1>  '••>  2p, 
be  a  primitive  scheme  of  periods,  and  let  f{zi,  •  •  • ,  Zn)  be  an 
integral  function  of  its  p  arguments.  If,  for  every  period  from 
this  scheme,  a  relation  of  the  form  holds: 

f(z,  H-  co,3,  •  •  •,  s,,  +  a;,^)  =  e^^^'VC^u  '',  ^p), 

where  Lp{z)  is  a  linear  (homogeneous  or  non-homogeneous,  but 
integral)  function  of  Zi,  •  •  • ,  Zp,  then/ is  called  a  Jacobian  function. 
The  Jacobian  functions  have  been  studied  at  length  in  two 
memoirs  by  Frobenius,*  and  in  a  paper  by  Wirtinger.f  A 
Jacobian  function  can  be  expressed  in  terms  of  theta  functions 
of  p  arguments. 

§  5.     AuTOMORPHic  Functions  of  Several  Variables 

The  brilliant  results  obtained  b}^  Klein  and  Poincare  in  the 
early  eighties,  in  their  researches  relating  to  the  automorphic 
functions  of  a  single  complex  variable  turned  the  attention  of 
mathematicians  towards  functions  of  several  complex  variables 
which  admit  a  discrete  group  of  linear  transformations  into 
themselves,  and  we  find  from  that  time  to  the  present  day  a 
steady  stream  of  papers  in  this  field. 

Here,  however,  at  the  very  threshold  of  the  subject,  two  types 
of  groups  present  themselves,  corresponding  on  the  one  hand 
formally  to  the  linear  transformations  of  projective  space: 

(U  (         I  0''^  +  b'y  +  C      a"x  +  h"y  +  c"  \ 

■^  V'y\ax-\-hy-\-c'        ax  +  hy  +  c     J' 

and  on  the  other,  to  those  of  the  space  of  analysis: 

*  Journ.  fur  Math.,  97  (1884),  p.  16  and  p.  188. 
t  Monalshcfle  fiir  Math.  u.  Phys.,  7  (1896),  p.  1. 


FUNCTIONS   OF   SEVERAL  COMPLEX   VARIABLES.  123 


(2) 


Y         I    ax+  b  ay  -\-  ^  \ 
V'y\    cx+d'      72/  +  5   /' 

/  I  a'y  +  b'  a'x  +  ^'\ 

X'^l  c'y+d"  y'x  +  b')' 


Hypermodular  Functions. — The  first  papers  to  appear  in  this 
field  dealt  with  groups  of  the  type  (1).  Picard*  began  by  in- 
vestigating a  class  of  functions  of  two  independent  variables 
analogous  to  the  elliptic  modular  functions.  It  is  a  familiar 
fact  that  a  hypergeometric  integral 


r 


dt 


}/t(t  -l){t-x)' 


where  g,  h  denote  any  two  of  the  four  points  0,  1,  oo,  x,  is  a 
solution  of  the  linear  differential  equation 

Let  ui,  C02  be  two  linearly  independent  solutions  of  this  equa- 
tion, and  set 

77= m- 

Then  the  equation 

f(x)  =  u 

defines  x  as  a  function  of  u,  and  this  function  is  analytic  through- 
out the  whole  upper  half  of  the  w-plane,  but  cannot  be  continued 
analytically  beyond  this  region. 

Picard  passes  to  analogous  functions  of  x,  y,  namely  those 
defined  by  one  of  the  integrals 


r 


dt 


,v 


Vt{t-  m-x)(t-y) 


where  g,  h  denote  any  two  of  the  five  points,  0,  1,  oo ,  x,  y.     These 
functions  satisfy  a  simultaneous  system  of  linear  partial  differ- 

*  C.  R.,  93  (1881),  p.  835;  ibid.,  94  (1882),  p.  579;  Acta,  2  (1883),  p.  114. 
Alezais,  "  Sur  une  classe  de  fonctions  hyperfuchsiennes,"  etc.,  Paris,  1901. 
10 


124  THE   MADISON   COLLOQUIUM. 

ential  equations  of  the  second  order,  the  coeflBcients  being 
polynomials  in  x  and  y,  at  most  of  the  third  degree,  with  integral 
coefficients. 

These  equations  admit  three  linearly  independent  solutions, 
coi,  C02,  C03.  If  the  latter  be  suitably  chosen  and  their  ratios  set 
equal  to  two  new  variables, 


CtJo 

Oiz 

-  w. 

—  x 

Oil 

Wl 

then  these  equations  deJSne  x  and  y  as  single-valued  functions  of 
w,  v.  The  domain  of  definition,  D,  is  that  part  of  the  four- 
dimensional  space  of  the  variables  u  =  u'  -\-  iu",  v  =  v'  -{■  iv", 
in  which 

2v'  +  u'^  +  u"^  <  0. 

The  proof  is  given  by  means  of  the  solution  of  Jacobi's  problem 
of  inversion  for  p  =  3;  cf.  §  2. 

Picard  shows  that  the  functions  thus  obtained  admit  a  properly 
discontinuous  group  of  linear  transformations  of  the  type  (1) 
which  carry  D  over  into  itself,  the  coefficients  being  of  the  form 
k  -\-  ZX,  where  k  and  I  are  integers,  and  X  is  a  complex  cube  root 
of  unity.  These  transformations  are  closely  related  to  those  of  a 
ternary  group: 

X=  Mix+P,y-\-R^z, 

Y=  M2X+P2y+R2Z, 

Z  =  Ahx  +  P,y  +  R^z, 

—  the  coefficients  here  being  also  rational  functions  of  X,  —  which 
leave  the  Hermiteian  form 

xx-{-  yy  -\-  zz 

unchanged,  where  x  denotes,  as  usual,  the  conjugate  of  x. 

Generalizations  of  Riemann's  P-Function.  The  investigations 
on  which  we  have  just  reported  suggest,  tlirough  the  hyper- 
geometric  integral  and  the  hypergeometric  differential  equation 


FUNCTIONS   OF   SEVEEAL   COMPLEX   VARIABLES.  125 

mentioned  at  the  outset,  Riemann's  researches  on  binary  families. 
In  fact,  Appell*  had  just  been  engaged  in  extending  these  results 
to  quaternary  families  of  functions  of  two  independent  variables, 
and  Picardf  had  himself  been  working  in  the  same  field. 

§  6.    Continuation.     Hyperfuchsian  and  Hyperabelian 

Functions 

A  further  paper  of  Picard|  deals  with  functions  F(u,  v)  mero- 
morphic  in  their  domain  of  definition,  D,  which  consists  of  the 
interior  of  the  hypersphere 

u'^  +  u"^  +  v'-  +  v"-  <  1, 

and  admitting  a  group  of  transformations  into  themselves  of 
type  (1).  The  fundamental  domain  of  the  group  lies  wholly 
within  D.  There  is  an  allied  system  of  simultaneous  linear 
partial  differential  equations  of  the  second  order. 

Between  three  such  functions  there  always  exists  an  algebraic 
relation,  —  a  property  corresponding  to  Weierstrass's  first  theorem 
concerning  periodic  functions  (§  4,  end),  and  these  functions 
serve  to  uniformize  such  an  algebraic  configuration. 

Double  integrals  on  the  corresponding  algebraic  configuration! 
are  studied,  being  uniformized  as  functions  of  u,  v,  and  in  this 
investigation  we  have  a  forerunner  of  Picard's  researches  on 
algebraic  functions  of  two  variables,  to  which  we  shall  presently 
turn. 

Functions  of  the  classes  hitherto  treated,  namely,  those  which 
admit  a  group  of  transformations  of  type  (1),  are  called  hyper- 
fuchsian functions. \\     The  definition  is  not  restricted  to  functions 

*  C.  R.,  90  (1880),  pp.  296,  731;  Journ.  de  Math.  (3),  8  (1882),  p.  173. 

tC.  R.,  90  (1880),  pp.  1118,  1267;  Ann.  Ec.  Norm.  (2),  10  (1881),  p.  305. 

t  C.  R.,  96  (1883),  p.  320;  C.  R.,  99  (1884),  p.  852.  We  note  here  a  paper  by 
Poincare,  C.  R.,  94  (1882),  p.  840,  in  which  automorphic  functions  of  two 
variables  are  obtained  from  the  theory  of  numbers.  Cf.  also  papers  by  Picard, 
Acta,  1  (1883),  p.  297;  ibid.,  5  (1884),  p.  121;  Ann.  Ec.  Norm.  (3),  2  (1885), 
p.  357. 

§  Cf.  Ill,  §  1. 

11  Picard,  Acta,  5  (1884),  p.  121. 


126  THE   MADISON   COLLOQUIUM. 

for  which  D  is  the  hypersphere,  but  includes  at  least  all  functions 
admitting  a  properly  discontinuous  group  of  type  (1)  and  mero- 
morphic  in  a  domain  D  defined  by  a  relation 

giti',  u",  /,  /')  <  0, 

where  gr  is  a  quadratic  polynomial.  Moreover,  the  functions 
cannot  be  continued  analytically  beyond  D. 

In  this  same  year  Picard*  began  the  investigation  of  functions 
which  admit  a  group  of  transformations  of  type  (2).  These 
functions  he  denoted  as  hyperabelian  functions,  since  the  first 
problem  which  he  was  led  to  study  concerning  them  was  one 
related  to  the  Abelian  thetas  and  the  Abelian  modular  functions, 
2?  =  2.  The  classes  discussed  yielded  functions  with  properties 
analogous  to  those  of  the  hyperfuchsian  functions. 

Generalizations.  In  a  systematic  development  of  the  theory 
of  the  automorphic  functions  of  several  complex  variables  a 
question  of  first  importance  is  that  of  the  existence  of  a  funda- 
mental domain  belonging  to  a  properly  discontinuous  group. 
A  solution  of  this  problem  for  such  groups  of  projective  trans- 
formations in  n  variables,  —  groups  of  type  (1),  —  has  been  given 
bv  Hurwitz.t 

The  extension  of  the  two  theorems  of  Weierstrass,  §  4,  for 
the  case  of  automorphic  functions  in  n  variables  has  been  treated 
by  Wirtingerf  by  the  aid  of  methods  of  the  general  theory  of 
functions. 

A  systematic  generalization  of  the  theory  of  a  class  of  hyper- 
abelian functions  was  outlined  by  Hilbert  and  elaborated  by 


*  Notes  in  the  Cotnptes  Rendus  for  1884,  followed  by  a  systematic  presen- 
tation in  Journ.  de  Math.  (4),  1  (1885),  p.  87.  Cf.  further  Bourget,  Toulouse 
Ann.,  12  (1898),  p.  D  1;  Humbert,  Journ.  de  Math.  (5),  5,  6,  7,  9,  10  (1899- 
1904),  and  (6),  2  (1906). 

^Malh.  Ann.,  61  (1905),  p.  325. 

tSitzungsber.  der  Wiener  Akad.,  108  (1899),  p.  1239.  For  the  special  case 
of  hyperabelian  functions  of  n  variables  cf.  Blumenthal,  Math.  Ann.,  56 
(1903),  p.  510;  ibid.,  58  (1904),  p.  497.  Picard  had  long  since  used  the  second 
theorem,  stated  for  automorphic  functions  of  two  variables;  cf.  Journ.  de 
Math.  (4),  1  (1885),  p.  313. 


FUNCTIONS   OF  SEVERAL   COMPLEX   VARIABLES.  127 

Blumenthal.*     The  group  is  that  in  which 

aixi+ ^i 

JiXi  -f-  Oi 

the  coefficients  being  taken  as  follows.  An  algebraic  domain  of 
rationality  is  assumed  as  given,  R  =  k,  where  k  denotes  a  root  of 
an  irreducible  algebraic  equation  in  the  natural  domain,  R  =  1. 
Furthermore,  all  the  roots  A-,  A-',  •  •  •,  /t^"~^^  shall  be  real.  The 
coefficients  ai,  •  •  •,  61  are  taken  in  R  =  k,  and  the  coefficients 
ai,  '  ■  •  ,bi  are  the  corresponding  numbers  of  the  domain  R=k''^~^\ 
Finally,  aidi  —  /3i7i  is  a  totally  positive  unit  of  the  domain 
R  =  k. 

The  subject  of  automorphic  groups  in  one  and  more  variables 
has  been  treated  systematically  by  Fubini.t 

§  7.    Algebraic  Functions  of  Two  Variables 

The  impetus  given  to  the  study  of  the  algebraic  plane  curves 
and  the  geometry  on  them,  through  the  researches  of  Pliicker, 
Cayley,  and  Clebsch,  in  connection  with  the  theory  of  the 
algebraic  functions  and  the  Abelian  integrals  as  developed  by 
Riemann,  early  made  itself  felt  in  the  study  of  algebraic  surfaces 
and  algebraic  functions  of  two  variables.  Thus  we  find  a  paper 
by  Clebsch  J  of  the  year  1868,  in  which  he  discovers  an  invariant 
of  an  algebraic  surface  analogous  to  the  deficiency  p  of  an  alge- 
braic curve.  The  latter  invariant  may  be  defined  as  the  number 
of  essential  constants  in  the  general  integral  of  the  first  kind,  i.  e., 
in  the  everywhere  finite  integral,  and  this  integral  can  be  written 
in  the  form 


/ 


*Cf.  preceding  reference.  Furthermore  Hecke,  Gottinger  Dissertation, 
1910. 

t  Introduzione  alia  teoria  dei  gruppi  discontinui  e  delle  funzioni  automorfe, 
1908. 

t  C.  R.,  67  (1868),  p.  1238.  Clebsch  had  only  the  adjoint  Q's  of  degree 
m  —  4.  The  everywhere  finite  double  integral  is  due  to  Noether,  Math.  Ann., 
2  (1870),  p.  293. 


128  THE   MADISON   COLLOQUIUM. 

where  f{x,y)  =  0  is  the  equation  of  the  ground  curve  Cm, 
assumed  irreducible,  and  Q{x,  y)  is  an  adjoint  polynomial  of 
degree  m  —  3.  If,  in  particular.  Cm  has  only  ordinary  double 
points,  Q  =  0  is  any  Cm-z  that  passes  through  these  points. 

Consider  now  an  irreducible  algebraic  surface  f{x,  y,z)  =  0  of 
degree  m  vfith.  only  ordinary  multiple  lines  and  isolated  multiple 
points.     Then  the  double  integral  (II,  §  2) 


// 


Q(x,  y,z) 

-ax  ay 


taken  over  an  arbitrary  regular  surface,  open  or  closed,  lying 
in  the  four-dimensional  Riemann  manifold  corresponding  to  the 
function  z  of  x,  y  defined  by  the  equation  /  =  0,  will  remain 
finite  provided  Q{x,  y,z)  =  0  is  an  adjoint  surface  of  degree 
m  —  4,  i.  e.,  a  surface  which  passes  through  the  multiple  lines  and 
has  a  multiple  line  of  order  A;  —  1  at  least  in  every  multiple  line 
of  /of  order  k;  and  which  moreover  has  a  multiple  point  of  order 
q  —  2  Sit  least  in  every  isolated  multiple  point  of  /  of  order  q* 
Such  an  integral  is  called  a  double  integral  of  the  first  kind.  The 
number  of  linearly  independent  integrals  of  this  class,  i.  e.,  the 
number  of  essential  constants  in  the  adjoint  polynomial  Q{x,  y,  z) 
is  called  the  deficiency,  or  more  precisely,  the  geometrical  de- 
ficiency,—  Flachengeschlecht,t  genre  geometrique,  —  in  distinc- 
tion from  the  numerical  deficiency  presently  to  be  considered,  and 
is  denoted  by  'pg.  It  is  an  invariant  under  the  group  of  birational 
transformations  of  the  surface: 


X  =  ri(.T,  y,  z), 
(A)  Y  =  r.ix,  y,  z), 

Z  =  Tsix,  y,  z),-^ 


x  =  Ri(X,  Y,Z), 
y  =  R,(X,  Y,  Z), 
z=  R,(X,  Y,Z).. 


In  case  the  surface  /  has  no  multiple  lines  or  points, 

(m-  l)(m-  2)(m-  3) 


Pa 


6 


*  Cf.  Pifard  et  Simart,  Fonctions  alg6briques  de  deux  variables,  vol.  1, 
1897,  ch.  7;  in  particular,  p.  189. 

t  The  invariant  is  due  to  Clebsch;  the  name  to  Noether. 


FUNCTIOXS    OF   SEVERAL  COMPLEX  VARLiBLES.  129 

The  Second  Deficiency.  There  is  a  second  numerical  invariant 
which  can  be  defined  as  follows.  Consider  the  linear  family  of 
adjoint  surfaces  of  degree  ??i  —  4: 

Q{x,  ?/,  s)  =  aiQi  +  0:2^2  +•••  +  oip^Qv,' 

These  surfaces  cut  the  ground  surface  /  =  0  in  certain  fixed 
curves,  —  including  always  the  multiple  curves  of  /,  —  and  a 
variable  curve,  I.  This  latter  curve  will,  in  general,  be  irreducible, 
and  we  assume  the  non-specialized  case.  It  is  a  twisted  space 
curve,  and  it  has,  as  such,  a  definite  deficiency,  which  can  be 
defined,  for  example,  as  the  deficiency  of  the  Riemann's  surface 
corresponding  to  the  curve.  This  deficiency  is  the  same  in 
general  for  the  different  curves  of  the  family,  and  it  is  this  number, 
2?^^^  which  is  called  the  second  or  numerical  deficiency,  —  Kur- 
vengeschlecht,*  le  second  genre.  It  is  an  invariant  under  the 
group  of  birational  transformations,  (^).t 

The  Line  Integral.  There  is  another  generalization  of  the 
Abelian  integrals  possible  for  the  algebraic  functions  of  two 
variables,  namely,1: 

SPdx+Qdy, 

where  P  and  Q  are  rational  functions  of  x,  y,  z,  the  third  variable 
being  a  root  of  the  irreducible  algebraic  equation  f{x,  y,  z)  =  0, 
and  where,  moreover,  the  condition  of  integrability  is  satisfied: 

dP  _  dQ 

dy        dx  ' 


*  This  invariant  is  due  to  Noether,  Math.  Ann.,  8  (1875),  p.  520. 

t  Cf.  Picard  et  Simart,  I.  c,  p.  206.  Noether  introduced  a  further  invariant, 
p^^^,  namely,  the  number  of  variable  points  of  intersection  of  two  curves  I. 
In  general,  p<^^  =  p^^>  —  1,  but  for  special  surfaces  p^^>  <  p'^'  —  1.  Cf. 
Picard  et  Simart,  ibid.,  p.  209. 

J  Picard,  Journ.  de  Math.  (4),  1  (1885),  p.  281;  ibid.  (4),  5  (1899),  p.  135. 
The  latter  paper  is  the  memoir  to  which  the  prize  of  the  Paris  Academy  of 
Sciences  was  awarded.  It  forms  the  foundation  of  the  later  presentation  of 
the  theory  of  Picard  and  Simart,  Fonctions  alg^briques  de  deux  variables, 
Paris,  1897-1900. 


130  THE   MADISON   COLLOQUIUM. 

Such  an  integral  is  a  function  of  two  independent  variables, 
and  these  may  be  taken  as  x,  y  or  y,  z  or  z,  a: 

A  division  of  such  integrals  into  three  classes,  corresponding 
to  the  three  classes  of  Abelian  integrals,  at  once  suggests  itself. 
In  the  first  paper  above  referred  to  Picard  studies  the  integrals 
of  the  first  class,  namely,  the  everywhere  finite  integrals.*  He 
finds  here  a  situation  diametrically  opposite  to  that  in  the  case 
of  the  Abelian  integrals.  If  f{x,  y)  =  0  is  an  irreducible  alge- 
braic equation  of  degree  greater  than  2,  there  will  in  general 
exist  integrals  of  the  first  class  corresponding  to  it;  it  is  only 
when  the  curve  is  highly  specialized  (unicursal)  that  this  is  not 
the  case. 

To  the  non-specialized  algebraic  surface  of  arbitrary  degree, 
however,  there  correspond  no  integrals  of  the  first  kind  with  the 
trivial  exception  of  a  constant.  A  special  class  of  surfaces  and 
integrals  is  treated,  the  former  being  those  which  can  be  uni- 
formized  by  means  of  quadruply  periodic  functions  of  two  inde- 
pendent variables. 

It  was  in  these  papers  that  Picard  began  the  study  of  questions 
relating  to  the  connectivity  of  the  surfaces  which  present  them- 
selves. The  points  of  an  algebraic  surface  fill  a  four-dimensional 
region,  —  be  that  region  assumed  as  a  four-dimensional  manifold 
in  space  of  six  or  more  dimensions,  or  as  a  multiple-sheeted 
Riemann  manifold,  or  as  a  fundamental  domain,  for  which  the 
parallelogram  of  periods  is  the  prototype.  In  this  four- 
dimensional  manifold  the  linear  cycles  (closed  curves)  and  the 
two-dimensional  cycles  (closed  surfaces)  are  of  especial  im- 
portance. Picard  finds  the  striking  result  that,  in  the  case  of  a 
non-specialized  algebraic  surface,  any  linear  cycle  can  be  drawn 
together  continuously  to  a  point.  This  fact  explains,  —  or  is 
explained  by,  —  the  non-existence  of  integrals  of  the  first  class 
on  such  a  surface. 

On  the  other  hand,  a  non-specialized  algebraic  surface  does 

*  Picard's  first  publication  relating  to  the  integrals  of  the  second  class 
appeared  in  the  Comples  Rendus,  100  (1885),  p.  843. 


FUNCTIONS   OF   SEVERAL  COMPLEX  VARL\BLES.  131 

admit  two-dimensional  cycles,  and  it  is  these  that  form  the 
analogue  of  the  linear  cycles  in  the  case  of  the  algebraic  functions 
of  a  single  variable.  With  these  are  connected  the  double 
integrals  of  Noether. 

The  methods  employed  in  these  early  geometric  investigations 
are  largely  those  of  intuition  and  analogy,  Picard  recognizes 
this  fact,  but  points  out  that  his  chief  object  was  to  throw  light 
on  a  theory  at  that  time  wholly  new. 

Geometry  on  Algebraic  Curves  and  Surfaces.  The  purely  al- 
gebraic theory  of  the  geometry  of  systems  of  points  on  algebraic 
curves  has  been  extended  to  algebraic  surfaces  and  systems  of 
curves  lying  on  them.* 

The  Point  of  View  of  the  Theory  of  Numbers.  The  methods  of 
the  theory  of  algebraic  numbers,  first  extended  to  the  algebraic 
functions  of  a  single  variable,  have  been  used  by  Henself  for 
the  study  of  algebraic  functions  of  two  variables.  In  his  treat- 
ment of  the  theory  of  the  algebraic  functions  of  a  single  variable 
Weierstrass  had  used  purely  algebraic  methods.  Hensel  de- 
scribes his  own  methods  for  algebraic  functions  of  two  variables 
as  the  direct  generalization  of  Weierstrass's  methods. 

In  a  preliminary  study  of  these  functions  Hensel  deduces 
series  developments  which  apply  to  the  neighborhood  of  a 
branch-line  or  of  a  multiple-line  of  the  surface.  The  form  of  the 
development  in  the  neighborhood  of  a  finite  point,  which  we  will 
take  as  the  origin  (0,  0,  0),  is  the  following: 

z  =  e^(xXy  -  yoY'"  +  e,(x){y  -  y,)"^  +  •  •  •, 

where  6  is  a  positive  integer.  The  coeflScients  ek{x)  and  the 
variable  yo  are  analytic  functions  of  ^,  where 

and  a  is  a  positive  integer.     In  fact,  the  equation  of  the  branch 

*  Xoether,  Math.  Ann.,  2  (1870),  p.  293;  ibid.,  3  (1871),  pp.  IGl,  547;  ibid., 
8  (1875),  p.  495.  Picard  et  Simart,  Fonctions  algebriques  de  deux  variables, 
vol.  2. 

t  Acta,  23  (1900),  p.  339;  Jahresber.  D.  M.-V.,  8  (1899),  p.  221. 


132  THE   MADISOX   COLLOQUIUM. 

of  the  discriminant  under  consideration  is 

§  8.    Analysis  Situs 

In  closing  we  refer  briefly  to  the  subject  of  analysis  situs  in 
the  geometry  of  n  dimensions.  Riemann  was  the  first  to  recognize 
the  importance  of  this  subject  for  the  surfaces  which  bear  his 
name.  He  had  also  thought  about  the  problem  for  higher 
manifolds.*  Bettif  considered  the  simple  closed  cycles  of  one 
dimension  (curves),  of  two  dimensions  (surfaces),  and,  generally, 
of  ??i-dimensions,  m  =  1,  2,  •  ■  • ,  n  —  1,  which  can  be  described 
in  the  ?i-dimensional  region  under  consideration,  and  he  intro- 
duced the  numbers  called  after  him,  which  indicate  how  many 
cycles  of  a  given  class  are  needed  as  a  basis  to  represent  a  general 
cycle  of  that  class. 

Attention  has  already  been  called  to  Picard's  work  on  questions 
in  this  field  relating  to  algebraic  surfaces,  §  7. 

Poincare  perceived  the  value  of  this  branch  of  geometry  for 
analysis  and  published  a  series  of  papers  on  the  subject.!  Fol- 
lowing Betti,  he  considered  integrals  extended  over  closed  m- 
dimensional  manifolds  (cycles)  in  the  n-dimensional  region,  and 
he  found  the  conditions  that  the  value  of  the  integrals  be 
invariant  of  a  restricted  deformation  of  the  manifold;  II,  §  2. 
Such  integrals  may  form  the  basis  for  determining  the  Betti 
numbers.  § 

*  Cf.  the  fragment  in  his  collected  works,  Werke,  1  ed.,  p.  448;  2d  ed.,  p.  479. 

t  Annali  di  mat.  (2),  4  (1870-71),  p.  140. 

J  Cf.,  in  particular,  Journ.  Ec.  Polytech.  (2),  Cah.  1  (1895),  p.  1;  also  the 
account  given  in  Picard  et  Simart,  Fonctions  algebriques  de  deux  variables, 
vol.  1,  ch.  2. 

§  An  elementary  geometric  treatment  of  the  analysis  situs  of  hypermanifolds 
has  recently  been  given  by  Veblen  and  Alexander,  Annals  of  Math.  (2),  14 
(1913),  p.  163. 


LECTURE  II 

SOME   GENERAL  THEOREMS 

§  1.     Definitions  and  Elementary  Theorems 
Let 

F{Zu    •'•,Zn) 

be  a  complex  function  of  the  n  complex  variables 

Z/t  =  Xk  -\-  iyk,  k  =  1,2,  ■•  -yTi, 

which  is  defined  uniquely  at  each  point  of  a  2?i-dimensional 
continuum  T.  Of  the  two  current  definitions  mentioned  in  I, 
§  1,  we  will  choose  the  second  and  say:  F  is  analytic  in  T  if,  at 
every  point  of  T,  it  admits  a  derivative  with  respect  to  each  of 
the  complex  arguments  Zi,  •  •  • ,  2„  and  if,  furthermore,  it  is 
continuous  in  T.  The  latter  condition  turns  out  to  be  a  con- 
sequence of  the  former,  cf.  §  5,  and  may,  therefore,  be  stricken 
from  the  definition.  But  it  is  better  to  retain  it  for  a  time,  since 
it  suffices  for  a  simple  proof  of  the  integral  theorems,  and  with 
the  aid  of  these  all  the  principal  theorems  are  readily  established. 

The  function  F(zi,  •  •  • ,  Zn)  is  said  to  be  analytic  in  a  point 
(ai,  •  •  • ,  a„)  if  it  is  analytic  throughout  some  region  T  containing 
the  point  in  its  interior.  Similarly,  F  is  said  to  be  analytic  in  a 
manifold  M  of  one  or  more  dimensions  if  it  is  analytic  throughout 
a  region  T  containing  M  in  its  interior.  If  M  is  closed,  i.  e,, 
if  M  contains  its  boundary  points,  then,  for  F  to  be  analytic  in 
M,  it  is  clearly  sufficient  that  F  be  analytic  in  every  point  of  M. 

The  Canchy-Riemann  Differential  Equations.  The  differential 
equations  which  the  real  part  u  (or  the  coefficient  v  of  the  pure 
imaginary  part)  of  an  analytic  function  satisfies  are  the  following: 


dxkdxi       dykdyi         '  dxkdyi      dykdx 

133 


134  THE   RLU)ISON   COLLOQUIUM. 

Thus,  when  n  =  2,  there  are  four  equations:* 

-u        c-u  c~ii^       d-u 


Cylindrical  and  B-Regions.  A  general  region  of  hyperspace 
can  be  described  analytically  by  one  or  more  inequalities.  Thus 
the  interior  of  the  hypersphere  of  radius  r,  with  its  centre  at  the 
origin,  is  given  by  the  inequality 

.Tl^  +  .T./  +  .  . .  +  xj  <  r\ 
or  by  the  pair  of  inequalities : 


2 


—    Vr^  —  X-^  —    •  •  •  —  Xm-\    <  Xm  <Vr-—  X-^  —    ••■   —  Xm~\ 

A  particularly  simple  and  important  class  of  regions  in  the 
space  of  the  n  complex  variables  Z\,  •  •  •  ,Zn  is  the  following.  Let 
Tk,  h  =  \,  •  •  -,  n,  be  an  arbitrarj^  two-dimensional  continuum 
in  the  complex  z^^-plane,  and  let  Zk  be  any  one  of  its  points. 
Then  the  region  of  2?z-dimensional  space  whose  points  (a'l,  yi, 
^2,  •  •  • ,  yn)  correspond  to  Zi,  •  •  • ,  z„  is  called  a  cylindrical  region, 
and  may  be  denoted  hy  (T)  =  (Ti,  •  •  • ,  Tn) ,  or,  more  simply, 
by  T.     It  is  a  continuum,  and  so  consists  only  of  interior  points. 

Let  Bk,  k  =  1,  •  •  •,  n,  be  a  regular  region  of  the  z^-plane,  i.  e., 
a  finite  continuum  plus  its  boundary,  the  latter  consisting  of  a 
finite  number  of  regular  curves  having  a  finite  number  of  multiple 
points  and  points  of  intersection;  and  let  Zk  be  any  point  of  Bk. 
Then  the  corresponding  region  of  the  2?i-dimensional  space  shall 
be  called  a  regular  cylindrical  region  or  a  B-region;  and  it  shall 
be  represented  as  {B)  =  {Bi,  •  ■  • ,  Bn),  or,  more  simply,  as  B. 

The  boundary  of  a  cylindrical  region  is  composed  of  those 
points  (21,  •  •  • ,  z„)  for  which  at  least  one  Zk  lies  on  the  boundary 
of  its  Tk  or  Bk.  It  consists  of  a  single  piece,  and  is  a  manifold 
of  2w  —  1  dimensions. 


*  TomcaT6,  C.  R.,  96  (1S83),  p.  238;  Ada,  2  (1883),  p.  99;  ibid.,  22  (1898), 
p.  112. 


FUNCTIONS    OF   SEVERAL   COMPLEX   VATIL\BLES.  135 

It  is  sometimes  useful  to  think  of  a  J5-region  as  a  rectangle, 
when  n  =  2,  or  as  a  parallelepiped,  when  n  =  3,  or  as  a  pris- 
matoid  in  higher  space,  just  as  we  picture  curves  and  surfaces 
to  ourselves  in  the  plane  or  in  space,  even  when  the  coordinates 
are  complex;  the  point  being  that  the  variation  to  which  each 
coordinate  is  subject  is  independent  of  that  to  which  any  other 
coordinate  is  subject. 

Cauchy's  Integral  Formula.  A  first  form  for  this  formula  is 
that  suggested  by  a  5-region.  Let  F{zi,  •  •  •,  z„)  be  analytic  in 
a  region  T,  and  let  5  be  a  regular  region  lying  in  T.  For  con- 
venience, let  ?i  =  2,  It  is  at  once  seen  that  F  is  given  for  any 
point  interior  to  B  by  the  formula: 


1 ^«^2, 

t2  —  22 


where  Ck  denotes  the  boundary  of  Bk  and  the  integral  is  extended 
in  the  positive  sense. 

This  iterated  integral  suggests  readily  a  double  integral, 
extended  over  a  surface,  i.  e.,  a  two-dimensional  manifold, 
which  lies  in  the  boundary  of  B.  But  the  complete  boundary  of 
5  is  a  2n  —  1  =  3-dimensional  manifold,  and  so  the  analogy 
with  Cauchy's  Integral  Formula  for  n  =  1: 

is  in  so  far  only  partial,  that  the  earlier  integral  (2)  is  extended 
over  the  complete  boundary  of  the  region  for  z,  whereas  the  present 
integral  (1)  is  extended  over  a  manifold  of  lower  order  which  lies 
in  the  boundary. 

§  2.    Line  and  Surface  Integrals,  Residues,  and  their 

Generalizations 

In  space  of  three  dimensions  the  surface  integral  of  a  contin- 
uous real  function,  f{x,  y,  z),  extended  over  a  curved  surface  S, 
is  defined  in  the  familiar  manner: 


136  THE  MADISON    COLLOQUIUM. 

I    1  f{x,  y,z)di:  =  lim  Z/(a'jt,  y^,  Zk)M:k, 

t/vc/  n=oo  A=l 

the  element  of  surface,  AS^,  being  taken  as  an  essentially  positive 
quantity. 

Allied  with  this  integral  is  the  other  surface  integral: 

I    \  A  dy dz  -{-  B dzdx  -\-  Cdxdy, 

where  A,  B,  C  are  continuous  functions  of  x,  y,  z.  Here,  the 
sign  to  be  attached  to  the  differential  factor,  dydz,  etc.,  requires 
special  definition,  and  can  be  assigned  in  terms  of  the  sense  of  an 
indicatrix  which  moves  continuously  over  the  surface,  or  in 
terms  of  the  signs  of  the  Jacobians 

"^{y,  z)  d{z,  x)  d{x,  y) 

dis,t)  '    ,     d{s,t)  '  dis,t)  ' 

where  s,  t  are  parameters  by  means  of  which  x,  y,  z  are  expressed. 

Such  integrals,  suitably  generalized  for  manifolds  of  order  k 
in  space  of  m  dimensions,  have  been  applied  by  Picard  and 
Poincare*  to  analytic  functions  of  n  complex  variables.  Poin- 
care  develops  conditions  that  the  value  of  such  an  integral, 
extended  over  a  closed  manifold,  be  invariant  of  slight  deforma- 
tions of  the  manifold,  and  hence  also  of  large  variations,  provided 
the  manifold  retains  its  character  and  does  not  sweep  over  a 
point  in  wliich  an  integrand  is  discontinuous  or  the  conditions 
in  question  cease  to  hold.     Cf.  also  I,  §  8. 

Residues.  The  value  of  any  such  integral  Poincare  calls  a 
residue.  Only  bilateral  manifolds  come  into  consideration, 
since  an  integral  extended  over  a  unilateral  manifold  evidently 
vanishes. 

As  a  first  application  of  the  foregoing,  consider  Cauchy's 
integral  formula.     In  the  form  in  which  it  stands  above,  the 

*  Picard,  C.  R.,  96  (1883),  p.  320;  ibid.,  a  scries  of  papers  in  vols.  102-3 
(1886).  Poincar6,  C.  It.,  102  (1886),  p.  202;  Ada,  9  (1887),  p.  321,  where 
reference  to  Jacobi  and  Marie  in  connection  with  these  integrals  is  made. 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES.  137 

integral  may  be  interpreted  as  a  residue,  the  particular  surface 
over  which  the  integration  is  extended  lying  in  the  boundary  of 
the  cylindrical  region  B.  It  appears,  however,  on  comparison 
with  Poincare's  criteria  that  the  surface  may  be  deformed  con- 
tinuously without  altering  the  value  of  the  integral,  and  hence 
we  are  led  to  a  generalization  of  the  integral  formula.  The 
integral  appears  as  a  residue,  the  surface  of  integration  being 
thought  of  as  a  closed  surface  which  is  linked  in  a  certain  way 
with  the  two  singular  surfaces 

(3)  ti  —  zi  =  0     and     to  —  Z2  =  0 

after  the  fashion  of  a  closed  curve  in  space  of  three  dimensions 
which  is  linked  with  certain  right  lines  of  that  space.     For,  the 
singular  surfaces  (3)  are  manifolds  of  order  2,  not  3,  in  space 
of  4  dimensions,  and  so  they  do  not  cut  that  space  in  two. 
Poincare  establishes  the  following  theorem.     If 


R{w,  z)  = 


H(w,  z) 


G{w,  z) 
is  a  rational  function  of  w,  z,  and  if 

j  J  R(iv,  z)dwdz 

is  extended  over  any  regular  closed  surface  which  has  no  point 
in  common  with  the  singular  manifold  G(iv,  z)  =  0,  then  the 
above  integral  can  be  evaluated  in  terms  of  the  moduli  of  peri- 
odicity of  Abelian  integrals  belonging  to  the  algebraic  configura- 
tion or  configurations 

G{w,  z)  =  0. 

He  also  considers  the  case  that  the  surface  meets  the  singular 
manifold,  and  obtains  here  an  evaluation  in  terms  of  Abelian 
integrals  with  variable  limits  of  integration. 

§  3.    The  Space  of  Analysis,  and  Other  Spaces 

The  space  of  analysis  can  be  defined  as  coextensive  with  the 
point  set  { (zi,  •  •  • ,  2n) } ,  where  each  one  of  the  complex  \ariables 


13S  THE   MADISON   COLLOQUIUM. 

Zk  ranges  over  its  extended  plane, — the  Neumann  sphere.  A 
point  of  that  space  lies  at  infinity  if  at  least  one  of  its  coordinates 
is  at  the  north  pole  of  its  sphere.  A  function  /(zi,  •  •  • ,  z„)  is 
said  to  be  continuous,  analytic,  or  meromorphic  at  a  point  of  the 
infinite  region  if,  when  each  coordinate  Zh  which  becomes  infinite 
is  replaced  by  a  new  point  by  means  of  the  transformation 


,  _  <XkZk-\-  (3k 


Oik      ^k 

7k     5k 


+  Q^         Jk^O, 


the  transformed  function  is  continuous,  analytic,  or  meromorphic, 
— as  the  case  may  be, — at  the  transformed  point. 

\^^^at  are  some  of  the  reasons  for  this  extension  of  proper 
(finite)  space?  First,  it  is  natural.  In  the  case  of  analytic 
functions  of  a  single  complex  variable  the  reasons  are  well  known 
why  it  is  desirable  to  extend  the  proper  Gauss  plane  by  a  single 
point, — the  point  co.  What  more  natural,  then,  than  to  take 
as  the  space  of  analytic  functions  of  n  complex  variables  the 
space  defined  by  the  n  spheres  of  the  individual  variables? 

But  this  reason  is  superficial.  It  is  formal.  The  real  object 
of  extending  proper  (finite)  space  at  all  is  to  secure  theorems 
which  include  among  their  hj'potheses  some  requirement  relating 
to  the  behavior  of  the  function  when  one  or  more  of  its  arguments 
become  infinite.  It  is  not  essential  that  ideal  elements, — points 
at  infinity,— he  introduced.  The  requirements  can  be  stated  in 
terms  of  a  transformation,  usually  linear,  though  not  necessarily 
projective,  applied  to  the  points  of  space  proper,  and  the  behavior 
of  the  transformed  function  in  the  neighborhood  of  a  point  or 
points  for  which  the  latter  function  is  not  defined. 

Thus  a  function  of  a  single  complex  variable,  f(z),  can  be 
defined  as  analytic  at  infinity  without  introducing  any  ideal 
element  whatever  if  we  proceed  in  either  one  of  the  following 
ways.  In  both  cases  we  shall  demand  that/(2)  be  analytic  outside 
of  a  certain  circle  in  the  2-plane,  and  finite  along  this  circle. 
And  now  we  require  further  either  (a)  that  f(z)  remain  finite  in 
the  above  region ;  or  (b)  that,  if  we  set 


FUNCTIONS   OF   SEVERAL   COMPLEX  VARIABLES.  139 

1 


-'  ' 


m  =  <p(z'), 


the  function  (p{z'),  which  is  not  defined  in  the  point  z'  =  0,  but 
is  analytic  in  the  rest  of  the  neighborhood  of  this  point,  shall 
have  a  removable  singularity  in  the  point  z'  =  0. 

Returning  to  functions  of  several  variables,  let  us  raise  again 
the  question,  why  introduce  the  space  of  analysis?  A  con- 
tribution toward  an  answer  to  this  question  is  to  be  found  in  the 
two  theorems  of  §  4,  below.  For  simplicity,  let  us  restrict  our- 
selves to  the  first  one.  This  theorem  is  not  true  if  our  hypothesis 
be  merely  that  the  function  shall  be  meromorphic  in  every  point 
of  finite  space.  Some  further  hypothesis  relating  to  its  behavior 
at  infinity,  or  to  the  behavior  of  the  function  when  subjected 
to  certain  transformations,  is  essential.  And  now  Weierstrass 
supplied  this  condition, — or  appears  to  have  done  so,  —  in  the 
way  indicated  above. 

But  is  this  the  only  way  in  which  this  end  can  be  attained 
without  doing  violence  to  simplicity  or  custom?  By  no  means, 
as  we  shall  presently  see. 

Projective  Space  arid  the  Space  of  the  Homogeneous  Variables. — 
The  space  most  familiar  to  the  geometers  is  projective  space,  and 
this  space  is  mapped  in  a  (1,  oo)-fold  manner  on  the  space  of 
72+1  homogeneous  variables  Xo,  Xi,  •  •  • ,  Xn.  This  latter  space 
is  the  whole  finite  space  whose  points  are  (.ro,  .Ti,  •  •  •,  .x'„),  where 
each  coordinate  ranges  over  its  whole  finite  Gauss  plane,  the 
one  point  (0,  0,  •  •  •,  0)  being  excluded.  We  will  speak  of  it  as 
the  space  of  the  homogeneous  coordinates. 

The  functions  considered  in  this  space  had  their  origin  in 
projective  space  (itself  but  an  amplification  of  an  ordinary 
finite  space),  and  are  homogeneous  in  the  n -\-  1  variables, — 
polynomials,  algebraic  functions,  and  such  transcendental 
functions  as  are  suggested  by  the  names  of  Aronhold,  Clebsch  and 
Gordan,  Klein,  and  their  school. 

Might  it  not  have  been  possible  to  choose  the  complementary 
hypothesis  in  Weierstrass's  theorem  is  §  4,  not  with  reference 
11 


140  THE   MADISON   COLLOQUIUM. 

to  the  space  of  analysis,  but  in  terms  of  projective  space?  More 
precisely,  we  should  demand,  as  before,  that  the  function  be 
meromorphic  (III,  §  2)  in  all  points  of  the  proper  space  of  the 
variables  (si,  •  •  • ,  Zn)  and  we  should  then  add  the  following 
hypothesis:  Let  a  transformation  of  the  type 


■'k 


ao  +  ai  Zi  +  0:2  22  +  •  •  •  +  ttn 

0  <  |q;i|  +  |a2|  +  •  •  •  +  \a-,\, 


be  performed;  and  let  (a|,  •  •  •,  o,',)  be  a  finite  point  not  corre- 
sponding to  any  finite  point  of  the  (si,  •  •  • ,  s„)  space.  Then, 
if  the  transformed  function  be  defined  in  each  removable  singu- 
larity of  the  neighborhood  of  (a'l,  •  •  •,  a',)  as  equal  to  the  limiting 
value  which  it  approaches  in  that  point,  the  thus  extended  function 
shall  be  analytic  or  meromorphic  in  (oj,  •  •  •,  «,'). 

Will  the  function  j{z\,  •  •  • ,  z^)  under  these  conditions  be 
rational?  The  answer  is  affirmative,*  And  the  corresponding 
theorem  holds  for  the  algebraic  case  of  §  4  also.  What  reason  is 
there,  then,  for  preferring  the  space  of  analysis  to  projective 
space  as  the  space  in  which  monogenic  analytic  configurations 
are  to  be  studied? 

The  answer  is  that,  so  far  as  these  two  fundamental  theorems 
are  concerned,  there  is  none.  We  shall  have  two  theories  of 
algebraic  functions  of  several  variables,  and  of  the  related 
transcendental  functions,  —  and,  more  generally,  of  monogenic 
analytic  functions,  —  according  as  we  extend  finite  space  in  the 
one  way  or  the  other.  Moreover,  when  n  >  2,  the  choice  is  still 
larger,  for  the  variables  may  then  be  divided  into  two  classes, 
those  of  one  class  being  transformed  projectively,  and  those  of 
the  other  class  spherically,  i.  e.,  so  that  the  Neumann  sphere  of 
each  variable  goes  over  into  a  Neumann  sphere  of  a  new  vari- 
able.    Thus,  if  z  is  defined  as  an  algebraic  function  of  x,  y  by 

*  Osgood,  Transactions  Amer.  Math.  Soc,  13  (1912),  p.  159. 


FUNCTIONS  OF  SEVERAL  COMPLEX  VARL\BLES.      141 

the  irreducible  equation  /(.r,  y,  z)  =  0,  it  is  possible  to  associate 
with  this  function  the  surface  in  projective  space  given  by  setting 


Xq                         Xq 

_  X3 
Xo' 

It  is,  however,  also  possible  to  put 

•Tl                          Xz 

X  =  — ,         y  =  —, 

Xq'            ^         Xo' 

Zi 
-^0 

and  still  again  to  set 


_  iEl 

Vl 

2l 

> 

y  =  —, 

2  =  — 

Xq 

yo 

Zo 

There  is  another  geometry  that  is  w^ell  known,  —  the  geometry 
of  reciprocal  radii,  or  the  geometry  of  inversion.  It  would,  of 
course,  be  a  proceeding  entirely  coordinate  with  that  which  has 
been  set  forth  above  to  extend  the  finite  space  of  n  complex 
variables  to  the  space  of  that  geometry. 

These  questions  could  not  arise  in  the  case  of  analytic  functions 
of  a  single  complex  variable,  for  there  the  infinite  region  of  pro- 
jective geometry,  the  geometry  of  inversion,  and  the  space  of 
analysis  are  the  same,  namely,  one  point. 

For  the  case  of  two  complex  variables,  the  infinite  region  of 
the  space  of  analysis  and  the  infinite  region  of  projective  geometry 
are  different,  and  moreover  the  space  of  analysis  and  projective 
space  can  no  longer  be  transformed  on  each  other  in  a  one-to-one 
manner  and  continuously.  But  the  space  of  analysis  is  trans- 
formable in  a  one-to-one  (but  non-real)  manner,  and  continuously, 
on  the  space  of  the  geometry  of  inversion.  When  the  number  of 
complex  variables  exceeds  two,  all  three  spaces  are  distinct.* 


*  For  a  detailed  treatment  of  these  questions  cf.  a  paper  by  Buchcr, 
Bulletin  Amer.  Math.  Soc.  (2),  20  (1914),  p.  185.  We  note  that  the  infinite 
region  of  the  space  of  analysis  consists  of  n  complex  (n  — l)-dimonsional 
manifolds  (hyperplanes)  which  have  as  their  sole  common  point  the  point 
(00,  00,  ...,  00). 


142  the  madison  colloquium. 

§  4.     Rational  and  Algebraic  Functions 
To  Weierstrass  is  due  the  theorem  that  a  function  of  n  complex 
variables  which  is  meromorphic  at  every  point  of  the  space  of 
analysis  is  a  rational  function.* 

Weierstrass  did  not  define  the  space  in  which  the  function  is 
considered.  He  said  "  im  ganzen  Gebiete  seiner  Verander- 
lichen."  It  appears,  however,  from  more  explicit  statements 
in  similar  cases  f  that  he  thought  of  each  variable  as  an  arbitrary 
point  of  its  extended  plane. 

A  similar  theorem  holds  for  algebraic  functions.  If  a  function 
of  n  complex  variables  is  finitely  multiple-valued  and  if,  in  the 
neighborhood  of  every  point  of  the  space  of  analysis,  the  values 
of  the  function  can  be  so  grouped  as  to  satisfy  one  or  more  alge- 
broid  relations, 

^ow"*  +  A^.io'^-^  + h  Am  =  0, 

where  the  ^'s  are  analytic  in  the  point  in  question,  —  and  to  be 
exhausted  in  said  neighborhood  by  these  systems,  —  then  the 
function  is  algebraic. 

§  5.     Sufficient  Conditions  that  a  Function  of  Several 
Complex  Variables  be  Analytic 

In  order  that  a  function  of  two  real  variables  be  analytic  it  is 
not  enough  that  the  function  be  analytic  in  each  variable  sepa- 
rately when  the  other  is  held  fast,  as  is  shown  by  the  example: 

fi^'  y)  =  ,.2  V  ,;i »  0  <  I  ,r  I  +  I  ?/ 1 ; 

'^  ~i  y 

m  0)  =  0, 

the  function  being  considered  in  the  neighborhood  of  the  origin. 
When,  however,  we  allow  the  variables  to  take  on  complex  values, 
the  case  stands  otherwise. 


*  Journ.  filr  Malh.,  86  (1880),  p.  5=Werke  2,  p.  129.     The  theorem  was 
I)iovon  by  Hurwitz,  Journ.  fur  Math.,  95  (1883),  p.  201. 
t  Cf.  for  example  Werke,  3,  p.  100,  7th  Une  from  end. 


FUNCTIONS    OF   SEVERAL   COMPLEX   VARIABLES.  143 

Theorem.  Let  /(.v,  y)  be  defined  throughout  a  cylindrical 
region  (S,  S'),  §  1.  Let/(.r,  6)  be  analytic  in  *S,  for  every  choice 
of  h  in  S';  b,  when  once  chosen,  to  be  held  fast.  Similarly,  let 
/(a,  y)  be  analytic  in  S',  a  being  any  point  of  S.  Then  /(.r,  y) 
is  analytic  in  the  two  independent  variables  x,  y  throughout 
the  region  {S,  S'). 

The  theorem  is  readily  proven  if  the  further  hypothesis  be 
added  that  the  function  remain  finite,  and  under  this  restriction 
in  sufficiently  general  for  many  of  the  cases  which  arise  in  prac- 
tice.* It  is,  however,  of  distinct  interest  to  know  that  the  more 
general  theorem  is  true.  This  latter  result  has  been  established 
by  Hartogs.j 

The  theorem  holds  for  functions  of  any  number  of  variables. 

Further  theorems  of  the  character  of  those  here  considered 
are  given  in  the  next  paragraph,  Theorems  A,  B. 

§  6.    Sufficient  Conditions  that  a  Function  be  Rational 

OR  Algebraic 

Hurwitz's  proof  of  Weierstrass's  theorem,  §  4,  yields  more  than 
is  contained  in  the  statement  of  that  theorem.  By  means  of  it 
the  following  theorems  can  be  established. 

Theorem  1.  If /(si,  •  •  • ,  Zn)  is  meromorphic  at  every  point  of 
the  coordinate  axes;  i.  e.,  in  each  of  the  points 

(0,  •••,0,  2,,  0,  •••,0),  /.•=  1,  ••-,//, 

where  the  variable  Zk  ranges  over  the  whole  extended  2A-plane, 
then  f{zi,  •  •  • ,  Zn)  is  a  rational  function  of  its  arguments. 

This  theorem  can  be  stated  in  the  following  form. 

Theorem  1'.  If /(zi,  •  •  •,  z,i)  is  meromorphic  in  each  of  those 
points  of  the  infinite  region  which  corresponds  to  any  ?i  —  1 
north  poles  combined  with  any  point  whatever  of  the  7ith  sphere, 
then  the  function  is  rational. 

A  special  case  of  this  theorem  is  the  following. 

*  This  theorem  was  proven  bj-^  the  author,  Math.  Ann.,  52  (1899),  p.  4ll2 
t  Math.  Ann.,  62  (1905),  p.  1.     Cf.  also  Osgood,  ibid.,  53  (1900),  p.  461. 


144  THE   MADISOX   COLLOQUIUM. 

Corollary.  If  f{zi,  •  •  • ,  z„)  is  meromorphic  in  every  point  of 
the  infinite  region  of  the  space  of  analysis,  then  /  is  a  rational 
function  of  all  its  arguments. 

These  theorems  readily  suggest  others,  in  which  the  word 
meromorphic  is  replaced  in  the  hypothesis  by  analytic;  the  con- 
clusion then  being  that  the  function  is  a  constant. 

Theorem  A.  If /(zi,  •  •  •,  Zn)  is  analytic  in  every  point  of  the 
coordinate  axes,  then  /  is  a  constant. 

The  manifold  M  consisting  of  the  coordinate  axes  is  perfect, 
and  hence  /  is  analytic  in  a  2?z-dimensional  region  T  enclosing 
the  axes.  It  is  possible,  in  particular,  to  choose  a  positive  number 
h  so  that  /  is  analytic  in  the  region 

|zi|  <  h,       "  -,     \zk-i\  <  h,     \zk+i\  <  h,     •  •  •,     |2„|  <  h, 

Zk  ranging  over  the  whole  extended  z^-plane;  k  =  1,  •  •  •,  n. 
Consider  /  in  the  region 


S:         \zk\  <  h,  k  =  1,  • '  -,  n. 

Let  (fli,  •••,«„)  be  a  point  of  this  region.     The  function 

/(ai,    •  ••,  On-l,  Zn) 

is  analytic  over  the  whole  extended  2„-plane.     Hence  it  is  a  con- 
stant.    Hence 


dZr. 


=  0 


in  the  point  (oi,  •  •  •,  «„).     But  this  was  any  point  of  2. 

It  appears,  then,  that 

df 
^^-0,  k==h-.-,n, 

and  from  this  fact  follows  the  truth  of  the  theorem. 

As  in  the  case  of  Theorem  1,  so  here  the  theorem  admits  an 
alternative  statement. 

Theorem  A'.  If /(zi,  •  •  •,  Zn)  is  analytic  in  those  points  of  the 
infinite  region  of  the  space  of  analysis  which  correspond  to  any 


FUNCTIONS    OF   SEVERAL   COMPLEX   VARIABLES.  145 

n  —  1  north  poles  combined  with  any  point  whatever  of  the 
7ith.  sphere,  then  /  is  a  constant. 

As  a  special  case  of  the  theorem  we  have  the 

Corollary.  If  /(zi,  •  •  • ,  z„)  is  analj'tic  in  every  point  of  the 
infinite  region  of  the  space  of  analysis,  then  /  is  a  constant. 

This  last  result  can  be  stated  in  a  form  wholly  independent  of 
any  assumption  regarding  the  infinite  region. 

Theorem  B.  If /(zi,  •  •  •,  s„)  is  analytic  at  all  finite  points  out- 
side a  fixed  hypersphere  :* 

G  <  .ir  +  yr  +  ^'2'+  •••  +  yn\ 

and  if  /  is  finite  in  this  region,  then  /  is  a  constant. 

Returning  now  to  theorems,  relating  to  rational  functions, 
we  have  the  following. 

Theorem  2.  If  f{z\,  •  ■  • ,  Zn)  is  a  rational  function  of  each 
individual  variable,  when  all  the  others  are  assigned  arbitrary 
values  in  the  neighborhood  of  a  certain  fixed  point  and  then  held 
fast,  then  /  is  rational  in  all  its  arguments. 

The  proof  of  Theorem  I  is  covered  by  Hurwitz's  reasoning,  and 
the  same  is  true  of  Theorem  II,  provided  the  additional  hypo- 
thesis is  made  that  the  function  be  analytic  in  all  its  arguments 
in  the  neighborhood  of  the  fixed  point  in  question.  In  practice, 
this  further  condition  appears  usually  to  be  fulfilled.  For  a 
proof  that  this  condition  is  a  consequence  of  the  others  I  am 
indebted  to  Professor  E.  E.  Levi. 

Both  theorems  can  be  extended  to  algebraic  functions,  the 
hypothesis  then  being  that  the  function  is  A^-valued,  and  that, 
moreover,  it  is  algebroid,  where  before  it  was  meromorphic. 

§  7.    On  the  Associated  Radii  of  Convergence  of  a  Power 

Series 
Let 

be  a  power  series  convergent  for  a  set  of  values  of  the  arguments, 
*  This  hypothesis  may  equally  well  be  written  in  the  form 

G  <  \zi\  +  ■••  +\Zn\. 


146  '  THE   MADISON   COLLOQUIUM. 

no  one  of  which  is  zero.*     A  set  of  positive  numbers  I'l,  ■  •  -yrn 
such  that  the  series  converges  when 


\xk\   <  Tk,  k  =   1,    '-',11, 

but  diverges  when 

\xk\  >  Tk,  k  =  \,  '  •  -,  n, 

is  called  a  set  of  associated  radii  of  convergence. 

The  numbers  ri,   •  •  • ,  r^  are  in  general  mutually  dependent 
on  each  other.     Thus  in  the  case  of  the  series 

T,Xl'X2''  = 


1  —  a*ia:2 
it  is  clear  that 

rir2  =  1. 

Geometric  Interpretation.  Geometrically  the  associated  radii 
of  convergence  may  be  interpreted  as  follows.  Denote  by  Tk  the 
circle  \xk\  <  Pk  and  by  T  the  2r^-dimensional  cylindrical  region 
T=  (ri,  •..,  Tn). 

Let/(a-i,  •  •  •,  Xn)  be  analytic  at  the  origin.     Then  the  p/t's  can 

be  so  chosen  that  T  lies  in  the  region  of  definition  of  the  element 

/(a-i,  •  •  • ,  a-„)  in  question.     And  now  let  the  p/t's  increase.     Any 

system  of  values 

Pk  =  fk,  k  =  1,  •  •  • ,  n, 

such  that  the  function  is  analytic  in  the  corresponding  region 
T  =  (Ti,  '  •  ■ ,  Tn),  but  no  one  of  the  T^'s  can  be  replaced  by  a 
larger  circle  without  diminishing  some  other  Tk  and  have  this 
properly  preserved,  is  a  system  of  associated  radii  of  convergence. 
Thus  we  may  picture  to  ourselves  a  variable  cylindrical  region 
T  in  the  domain  of  definition  of  the  monogenic  analytic  function 
/(.Ti,  •  •  •,  Xn).  Those  regions  T  that  reach  out  to  singular  points 
of  the  function  and,  moreover,  are  maximum  regions  in  this 

*  By  a  convergent  multiple  series  Sw,,i ...  ^„  we  mean  a  series  such  that  every 
simple  series  formed  from  its  terms  converges.  If,  then,  a  multiple  series 
converges,  it  necessarily  converges  absolutely. 

Other  multiple  series  have  been  investigated  in  recent  years  by  Pringsheim 
and  Hartogs. 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES.  147 

sense,  that  they  cannot  be  expanded  for  any  one  of  the  variables 
Xk  without  being  contracted  for  others,  yield  associated  radii  of 
convergence. 

Detailed  Consideration  of  the  Case  n=2.  The  mutual  relation's 
between  the  r's  have  been  studied  extensively.  Let  the  number 
of  variables  be  two,  and  let  the  series  be  written : 

(1)  T.Cm.nX'^y^ 

If,  for  a  pair  of  values  a*o,  yo,  neither  of  which  is  zero,  the  terms 
of  the  series  (1)  remain  finite,  then  it  is  well  known  that  the  series 
converges  (and  hence  converges  absolutely),  when 

I  •'^'  I  <  '•.     I  ^  I  <  *> 

where  r  =  \xo\,  s  =  \yo\. 

Let  r  have  an  arbitrary  value  in  the  interval  0  <  ^  <  |a:o|. 
To  this  value  of  r  may  correspond  larger  values  of  5,  — in  fact, 
there  may  be  no  limit  to  s.  If,  however,  the  latter  is  not  the 
case,  let  <p(r)  denote  the  upper  limit  of  6'  for  the  value  of  r  in 
question.  Then  r  and  ip{r)  form  a  pair  of  associated  radii  of 
convergence.  Also,  (p{r)  is  spoken  of  as  the  associated  radius  of 
convergence  (i.  e.,  associated  with  r  as  independent  variable). 
If,  for  a  given  r,  s  has  no  upper  limit,  the  associated  radius  of 
convergence  is  said  to  be  infinite  * 

A  necessary  and  sufficient  condition  that  r,  s  be  associated  radii 
of  convergence  has  been  obtained  by  Lemaire,t  who  generalized 
a  familiar  theorem  of  Cauchy's  for  power  series  in  a  single 
variable.  It  is  as  follows.  Consider  the  points  of  condensation 
of  the  set  of  numbers 

where  m,  n  independently  range  over  the  positi\e  integers  and 
zero.  Then  the  condition  is  that  the  points  of  the  original  set 
remain  in  the  finite  region,  and  that  the  point  of  the  derived  set 
most  remote  from  the  point  0  be  situated  at  1. 

*  This  is  not,  of  course,  the  same  thing  as  saying  that,  for  such  a  value  of 
r,  (p{r),  becomes  infinite. 

t  Bull,  des  Sci.  Math.  (2),  20  (1896),  p.  2S6. 


148  THE   MADISON   COLLOQUIUM. 

The  corresponding  condition  holds  for  a  power  series  in  any 
number  of  variables. 

The  properties  of  the  function 

s  =  <p(r) 

have  been  investigated,  the  most  important  of  the  results  being 
the  following.*     First,  some  obvious  properties. 

If,  for  /'o  >  0,  the  associated  radius  of  convergence  is  infinite, 
then  it  is  infinite  for  every  smaller  value  of  r :  0  ^  r  ^  ro. 

As  r  increases,  (p{r)  decreases  or  remains  constant;  i.  e.,  (p{r) 
is  a  decreasing  monotonic  function  of  r.f 

The  domain  of  definition  of  (p{r)  consists  of  an  interval 

0  ^  i?o  <  r  <  i?i     or     0  ^  i?o  <  /•  <  ^ , 

where,  however,  it  is  not  obvious  whether  an  extremity  of  the 
interval  shall  pertain  to  the  interval  or  not. 

The  basal  theorem  relating  to  (p(r)  is  the  following. 

Theorem. X    Let 

be  a  double  power  series,  and  let 

s  =  (fir), 
O^Ro<r<IU,     resp.     0  ^  Ro  <  r  <  oo , 


*The  leading  results  here  given  were  obtained  by  Phragm6n  as  early  as 
18S3  and  published  by  him  in  a  notable  paper  cited  below.  They  were  extended 
to  n  variables  by  A.  Meyer,  Thesis,  Upsala,  1887. 

t  This  property,  together  with  the  property  that  <p{r)  is  continuous,  was 
given  by  Weierstrass  in  his  lectures;  W.  S.,  1880/81.  Cf.  also  the  next  refer- 
ence. 

t  This  theorem  in  its  present  form  was  first  given  by  Fabry,  C.  R.,  134 
(1902),  pp.  1190,  and  rediscovered  by  Hartogs,  Thesis,  Munich,  1904  and 
Habilitationsschrift,  1904:  =  Math.  Ayin.,  62  (1905),  p.  49  and  p.  81.  By  the 
aid  of  it  the  theorems  which  follow  in  the  text  are  easily  proven;  cf.  Fabry  and 
Hartogs,  1.  c. 

A  third  paper  closely  related  to  the  two  just  cited  and  containing  a  number 
of  their  results,  obtained  by  a  different  method,  was  published  by  Faber, 
Malh.  Ann.,  01  (1905),  p.  289.  Faber's  methods  apply  to  power  series  ^-ith 
ati}-  number  of  variables,  and  his  paper  contains  generaUzations  of  theorems 
discussed  in  the  text  for  the  case  of  two  variables. 


FUNCTIONS   OF   SEVERAL  C0:MPLEX  VAEIABLES. 


149 


be  the  function  which  corresponds  to  r  as  associated  radius.     Let 

r\  <  r-i  <  rs 
be  three  points  of  the  interval  of  definition  of  ^(r).     Then 

1     log  7-1     log  <p{ri) 

(2)  1     log  r2     log  (p{r2)     <  0. 

1     log  rz     log  ^(rs) 

This  is  the  relation  designated  by  Hartogs  as  the  Fundamental 
Property.  It  was  proven  by  Fabry  by  means  of  Lemaire's  theorem 
cited  above.  Hartogs  gave  several  proofs,  one  of  which  is 
based  on  his  function  Rx  defined  below.  He  has  also  thrown 
Fabry's  proof  into  exceedingly  simple  form.* 

The  theorem  admits  the  following  interpretation.     Let 

(3)  X  =  log  r,     y  =  log  s. 
Then 

1 1     ^'i    2/1 

(4)  1      X2      1/2      <  0. 

1       .T3      2/3 

Hence  the  curve  that  represents  y  as  a  function  of  x,  the  above 
curve  (3),  or 

(5)  y  =  co{x), 

is  always  continuous f  and  concave  downward. 

Let  the  function 

fix,y)  =  ZC;„.  „.!•'"  2/" 

be  analytic  in  the  points  {x,  0),  where  \x\  <  R.  If  the  associated 
radius  corresponding  to  one  single  value  of  r,  r  =  ri,  in  the  inter  • 
val  0  <  r  <  R'ls  infinite,  then  the  associated  radius  is  infinite  for 
every  value  of  r  in  this  interval.  In  this  case,  then,  the  function 
(p{r)  does  not  exist. 

*  Jahresber.  D.  M.-V.,  16  (1907),  p.  232. 

t  This  property  was  established  by  A.  Meyer,  Stockholm  Ved.-Ak.  Fork.  Ofv., 
40  (1883),  No.  9,  p.  15,  and  Phragmen,  ibid.,  No.  10,  p.  17.  Cf,  the  reference 
to  Weierstrass,  p.  148. 


150  THE   MADISOX   COLLOQUIUM. 

From  this  theorem  it  follows  that,  when  the  function  <f{r) 
exists  at  all,  its  interval  of  definition  reaches  back  to  the  origin: 
0  ^  r  <  i?. 

If  the  function  (p{r)  exists  and  is  constant  in  a  portion  of  the 
interval  of  definition,  then  ip{r)  is  constant  clear  back  to  the 
beginning  of  the  interval. 

The  function  co(.r)  possesses  a  finite  forward  derivative  and  a 
finite  backward  derivative,  neither  of  which  is  positive.  The 
same  is  true  of  the  function  s  =  (p{r). 

If  Xi  and  X2  are  any  two  points  of  the  interval  of  definition  of 

co(.r) : 

—  GO  <  xi  <  .v-2  <  log  R, 

neither  of  the  above-named  derivatives  in  the  point  .r2  exceeds 
either  one  of  the  derivatives  in  xi. 

From  these  results  it  is  clear  that,  if  P  is  any  point  of  the 
curve  (5),  then  a  straight  line  whose  slope  is  negative  or  nil  can  be 
drawn  through  P,  such  that  the  curve  nowhere  rises  above  the  line. 

By  means  of  such  lines,  —  "  tangents,"  as  Hartogs  calls  them, 
—  Hartogs  and  Faber*  show  that  the  fundamental  property  (2) 
is  the  only  condition  which  the  function  c?(r)  must  fulfil.  In 
other  words:  Let  (p{r)  be  anj- function  of  r  which  is  defined  through- 
out an  arbitrary-  interval  0  ^  r  <  i?,  is  positive  there,  and  is 
subject  to  the  condition  (2).  Then  there  exists  a  double  power 
series 

to  which  the  numbers  r,  s  correspond  as  associated  radii  of 

convergence,  where 

s  =  <p{r),  0  ^r<  II 

§  8.    Hartogs's  Function  R^ 
In  a  number  of  his  investigations  Hartogs  makes  extended 
use  of  a  function  R-c  which  can  be  defined  as  follows. f     Let 

*  Hartogs,  Malh.  Ann.,  62  (1905),  p.  84.  Faber,  1.  c.  Since  Faber  does  not 
introduce  the  logarithm,  the  tangents  appear  as  his  TF-curves. 

t  Hartogs,  Dissertation,  and  Malh.  Ann.,  62  (1905),  pp.  24,  25.  The 
notation  there  used  is  R'^o- 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES,  151 

f{x,  y)  be  analytic  in  the  point  (.I'o,  0) ;  let 

fix,  y)  =  llCm,n{x  -  .To)"*^"; 

and  let  r,  <p{r)  be  the  associated  radii  of  convergence  corresponding 
to  this  series,  if  they  exist.     Then 

Rx^  =  Vim  <p{r). 

1=0 

Thus  Rxo  is  defined  for  an  arbitrary  point  (.ro,  0) ;  and  if  this 
point  is  designated  merely  as  (.r,  0),  we  write,  as  a  matter  of 
notation,  simply  Rx. 

The  function  Rx  can  also  be  obtained  as  follows.  The  function 
f{x,  y),  analytic  in  {xq,  0),  is  analytic  at  all  points  (.ro,  y)  for 
which  1 2/ 1  is  duly  restricted.  The  upper  limit  of  the  radius  of 
the  latter  circle,  if  one  exists,  is  the  number  Rx^. 

The  geometric  interpretation  of  a  pair  of  complex  numbers 
as  a  point  of  the  plane  of  analytic  geometry  can  here,  too,  be 
used  with  advantage.  The  series  will  then  be  thought  of  as 
converging  throughout  a  certain  rectangle  with  its  centre  at 
(a*o,  0)  and  its  sides  parallel  to  the  coordinate  axes;  such  a 
rectangle  to  be  a  maximum  rectangle  in  the  sense  that  neither 
side  may  be  increased  without  diminishing  the  other  side.  And 
now,  as  the  base  parallel  to  the  .T-axis  approaches  0,  the  half- 
altitude,  if  it  remains  finite,  approaches  as  a  limit  Rx^. 

Or,  again,  we  may  use  the  geometric  interpretation  in  terms 
of  the  cylindrical  regions  of  §  1. 

It  is  obvious  that,  corresponding  to  any  arbitrary  point  .ro 
for  which  Rx^  exists,  the  function  /(a-,  y)  has  a  singular  point  in 
some  point  (.ro,  t/q)  for  which  [i/o|  =  Rx^. 

The  function  Rx  is  real  and  positive,  and  it  is  semi-continuous 
in  this  sense.  Let  .r  =  .ro  be  anj-  point  in  which  it  is  defined,  and 
let  c  be  an  arbitrarily  small  positive  number.  Then  there  exists 
a  positive  5  such  that 

^x  ^  Rxo  —  €,  |.r  —  .ro|  <  6. 

Let 

Z^^m.nX     y 


152  THE   MADISON    COLLOQUIUM. 

be  a  double  power  series  with  the  associated  radii  of  convergence 
r,  (p(r).  Consider  the  points  x  of  the  circle  \x\  <  r,  where  r 
has  an  arbitrary  value  in  the  interval  of  definition  of  the  function 
(p(r).  Then  the  lower  limit  of  Rx  for  the  points  of  this  circle  is 
equal  precisely  to  (p(r). 

On  the  following  theorem  Hartogs  bases  his  proof  of  a  number 
of  important  theorems.* 

Theorem.  Let  T  be  an  arbitrary  domain  of  the  .i-plane,  and 
let  f{x,  y)  be  analytic  in  the  points  {x,  0) ,  where  x  lies  in  T. 

Let  5  be  a  regular  region  lying  within  T,  and  let  p^  be  a  positive 
real  function  of  x,  such  that 

(1)  log  yx  is  harmonic  within  B: 

A  log  i)x  =  0; 

(2)  px  is  continuous  on  the  boundary  C  of  B,  and  the  boundary 
values  px\(,  are  positive. 

If,  now, 

then,  throughout  the  whole  interior  of  B, 

Bx  ^  l^x . 


Finally,  if  at  a  single  interior  point  the  lower  sign  holds,  then 
Bx  =  Px  throughout  B. 

Hartogs  finds  further  that  if  Bx  is  continuous  together  with 
its  partial  derivatives  of  the  first  and  second  orders,  then  Bx 
satisfies  the  differential  inequality: 

A  log  Bx  ^  0. 

The  definition  and  the  properties  of  the  function  Bx  here  con- 
sidered have  been  extended  to  the  case  that  f{x,  y)  is  allowed  to 
be  meromorphic  instead  of  being  restricted  to  being  analytic, 
in  a  paper  by  Levi.f 


*  Math.  Ann.,  62  (1905),  p.  40. 

t  E.  E.  Levi,  Ann.  di  mat.  (3),  17  (1910),  p.  12. 


functions  of  several  complex  variables.  153 

§  9.    On  the  Analytic  Continuation  of  a  Logarithmic 

Potential 

Let 

10  =  f{z) 

be  an  analytic  function  of  the  single  complex  variable  z,  and  let 

w  =  u  -\-  V  i,     z  =  X  -]-  y  i. 
Then 

u  =  <p{x,  y) 

is  a  logarithmic  potential  function  of  the  real  variables  .r,  y. 
Moreover,  as  is  well  known,  u  is  an  analytic  function  of  .r,  y. 
As  such,  it  admits  definition  for  complex  values  of  the  arguments, 
and  thus  gives  rise  to  a  monogenic  analytic  configuration  in  two 
independent  variables. 

For  the  real  values  of  x  and  y  for  which  the  logarithmic  potential 
was  originally  considered,  u  is  real.  It  is,  now,  quite  conceivable 
that  we  may  be  able  to  pass  continuously,  i.  e.,  by  analytic 
continuation  through  the  complex  domain,  to  another  part  of 
the  analj-tic  configuration  in  which  (.r,  y),  and  also  u,  are  real, 
and  thus  arrive  at  a  new  real  solution  Wi  of  Laplace's  equation, 
Aw  =  0,  not  obtainable  from  the  earlier  one  by  analytic  con- 
tinuation along  a  real  path.  1\\  particular,  the  function  f{z) 
may  have  a  lacunary  space,  and  it  is  conceivable  that  Uy  might  be 
defined  in  that  space. 

Study*  has  considered  this  question,  and  has  shown  that  the 
answer  is  negative.  The  only  real  solutions  of  Laplace's  equation 
which  can  be  obtained  by  analytic  continuation  are  those  which 
are  obtainable  by  continuation  along  a  path  lying  wholly  in  the 
real  domain. 

Study  generalizes  the  question  here  considered  and  solves 
the  corresponding  problem,  referring  at  the  same  time  to  a  paper 
of  Segre.f 


*  Math.  Ann.,  63  (1906),  p.  240. 
t  Ibid.,  40  (1892),  p.  465,  11.  10-14. 


15-i  the  madison  colloquium. 

§  10.    The  Representation  of  Certain  Meromorphic 
Functions  as  Quotients 

In  his  noted  memoir  of  1S7G  Weierstrass  showed  that  any 
function  of  a  single  complex  variable,  which  has  no  other  singu- 
larities than  poles  in  the  finite  region  of  the  plane,  can  be  ex- 
pressed as  the  quotient  of  two  integral  (rational  or  transcendental) 
functions. 

The  theorem  was  later  extended  by  Mittag-Leffler  to  the  case 
of  an  arbitrary  region.  A  function  meromorphic  in  such  a  region 
can  be  expressed  as  the  quotient  of  two  functions  each  analytic 
in  the  region.  In  both  cases,  the  numerator  function  and  the 
denominator  function  never  vanish  at  the  same  point  of  the 
region. 

Furthermore,  the  region  may  be  any  continuum  whatever, 
and  both  the  zeros  and  the  poles  may  be  chosen  arbitrarily  in 
it.  There  will  always  exist  a  function  with  the  given  zeros  and 
poles  and  otherwise  analytic  and  different  from  zero  in  the  given 
region. 

The  first  of  these  theorems  admits  generalization  for  a  function 
of  several  variables.  If  /(zi,  •  •  • ,  z„)  is  meromorphic  at  every 
point  of  finite  space,  then  there  exist  two  (rational  or  transcen- 
dental) integral  functions  G{zi,  -  •  • ,  Zn)  and  H{zi,  •  •  • ,  s„)  such 
that 

,,  .    _   H(Zl,    '■•,Zn) 

J(2l,    •  •  •,  Zn)   —  JTTZ  TT  • 

Moreover,  at  any  point  at  which  G  and  //  both  vanish,  the 
representation  is  a  normal  one;  i.  e.,  G  and  H  have  no  common 
factor  in  this  point;  IV,  §  1. 

This  theorem  was  stated  by  Poincare  for  the  case  of  two  vari- 
ables, and  he  gave  a  proof  based  on  harmonic  functions  in  four- 
dimensional  space.*  A  more  elementary  proof,  which  applies, 
moreover,  to  the  general  case  of  7i  variables,  was  later  published 
by  Cousin,  t 

*  Ada  Math.,  2  (1883),  p.  97. 
t  Acto  Mulh.,  19  (1895),  p.  1. 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES.  155 

In  addition,  Cousin  establishes  the  general  existence  theorem 
for  this  case,  namely,  that  the  zeros  and  the  singularities  may  be 
chosen  at  pleasure.  More  precisely,  this  condition  is  as  follows. 
To  each  point  (a)  =  (oi,  •  •  •,  a„)  of  finite  space  shall  be  assigned 
a  definite  region  ^(o)  including  this  point  in  its  interior,  and  a 
function 

J,       ,  V         H(a){Zi,    •  •  • ,  Zn) 

where  G^a)  and  H(^a)  are  both  analytic  in  T^a)  and  where,  in  case 
both  functions  vanish  at  the  same  point  of  T^a),  they  have  no 
common  factor  there.*  When  two  regions  T^a)  and  T^b)  overlap, 
the  corresponding  functions  /(a) (21,  •  •  •,  Zn)  and  /(b) (zi,  •  •  •,  2„) 
shall  be  equivalent  in  the  common  region,  i.  e.,  their  quotient, 
taken  either  way,  shall  remain  finite,  and  so  shall  have  at  most 
removable  singularities  there. 

Under  these  hypotheses  there  exist  two  integral  functions, 
G{zi,  •  ",  Zn),  H{zi,  '  • ',  Zn),  such  that  their  quotient 

J..  s        ^(21,  ■  •  -,  ^n) 

/(^"••••^«)  =  Gfe,  ...,z„) 

is  equivalent  to  f{a)(zi,  •  •  -,  Zn)  in  the  region  T(a)  for  all  values 
of  (a)  and  that,  at  all  points  at  which  G  vanishes,  this  quotient 
is  in  normal  form. 

From  the  theorems  of  the  next  paragraph  it  appears  that  both 
numerator  and  denominator  can  be  written  as  the  (finite  or 
infinite)  product  of  prime  factors. 

But  Cousin's  methods  extend  far  beyond  the  scope  of  this  case. 
Cousin  states  the  following  theorem. f  Let  S  =  (Si,  ••-,  Sn) 
be  an  arbitrary  cylindrical  region.  To  each  interior  point 
(a)  =  («!,  •  ■  •,  a„)  let  a  region  T(^a)  lying  in  S  and  including  (a) 
in  its  interior,  and  a  function /(a) (zi,  •••,  Zn)  analytic  in  (a) 
be  given.     When  two  regions  T(a)  and  T^b)  overlap,  the  corre- 

*  Cf .  IV,  §1.     The  denominator  function  G(a)(zi,  •••,  Zn)  will,  of  course, 
in  general  not  vanish  at  all,  and  in  that  case  can  be  set  =1. 
t  L.  c,  p.  60,  Theorems  XIII,  XIV. 

12 


156  THE  MADISON   COLLOQUIUM. 

spending  functions  shall  be  equivalent  in  the  common  domain. 
Then  there  exists  a  function  f(zi,  •  •  • ,  Zn)  analytic  in  5  and 
equivalent  to/(a)(2;i,  •  •  •,  z„)  in  T(,i)  for  all  points  (a)  of  S. 

This  theorem  carries  with  it  the  other  one,  in  which  the  word 
analytic  is  replaced  by  meromorphic,  and,  in  the  conclusion  the 
function  /  is  expressed  as  a  quotient : 

.,  .    _  H(Zi,    •  •  • ,  2n) 

m,    •••'^n^-(?(,^,    •••,2n)' 

which,  at  any  point  (a)  in  which  both  numerator  and  denominator 
vanish,  is  in  reduced  form. 

Dr.  Gronwall*  has  just  shown  by  an  example  that  the  theorem 
in  this  degree  of  generality  is  not  true.  It  is  true  if  the  region  »S 
is  simply  connected,  i.  e.,  if  each  region  Sk  is  simply  connected. 
One  of  these  regions,  Sk,  however,  may  be  multiply  connected. 

§  11.    IxTEGEAL  Functions  as  Products  of  Prime  Factors 

A  further  theorem  which  Weierstrass  established  in  the  memoir 
of  1876  is  this.  If  G(z)  be  any  integral  function  which  does  not 
vanish  identically,  but  which  has  an  infinite  number  of  roots, 
then  G{z)  can  be  written  as  an  infinite  product  of  prime  functions: 


where 


G(z)  =  T(z)  2"*  n  (  1  -  —  )  e<'"^'\ 

n=l  \  (1)1  / 

z        1  f  z  \-  l/~  \"" 


and  T{z)  is  an  integral  function  having  no  roots. 

As  has  already  been  pointed  out,  the  zeros  may  be  chosen  at 
pleasure. 

To  extend  this  theorem  to  integral  functions  of  several  complex 
variables  it  is  necessary  first  of  all  to  define  a  prime  function. 

An  integral  function  G{zi,  •  •  • ,  Zn),  which  vanishes  for  some 
point,  is  said  to  be  prime  or  irreducible  if  it  is  not  possible  to 

♦  Bvll.  Amer.  Math.  Soc.  (2)  20  (1914),  p.  173. 


FUNCTIONS  OF  SEVERAL  COMPLEX  VARIABLES.      157 

write  it  as  the  product  of  two  integral  functions: 

G{Zi,    •  ■  ■  ,  Zn)   =    Gi{Zi,    •  •  •  ,  Zn)G2  (Zl,    '  *  ',  2„), 

both  of  which  vanish.* 

The  roots  of  a  prime  function  yield  the  coordinates  of  all 
finite  points  of  a  certain  monogenic  analytic  configuration. 

Let  G{zi,  •  •  • ,  z„)  vanish  in  a  point,  but  not  vanish  identically. 

Then  the  equation 

G(zi,  -  • ' ,  Zji)  =  0 

defines  one  or  more  monogenic  analytic  configurations.  Let  M 
denote  one  of  them.  By  the  aid  of  Cousin's  theorem  it  is  possible 
to  infer  the  existence  of  an  integral  function  which  vanishes  in 
the  points  of  M  and  nowhere  else,  and  which,  moreover,  is  prime.f 
G{zi,  •  •  • ,  z„)  is  divisible  by  this  function. 

From  Weierstrass's  factor  theorem,  IV,  §  1,  it  now  follows 
that,  in  the  neighborhood  of  a  point  and  hence  throughout  any 
finite  region  of  2w-dimensional  space,  an  integral  function  which 
vanishes  there,  but  does  not  vanish  identically,  can  be  written 
as  the  product  of  a  finite  number  of  factors,  each  irreducible  in 
the  point  or  in  the  region,  multiplied  by  another  integral  function 
which  does  not  vanish  there. 

It  is  now  an  easy  matter,  by  the  methods  used  in  the  proofs  of 
Weierstrass's  and  Mittag-Leffler's  theorems,  to  establish  the 
proposed  generalization:  An  integral  function  which  vanishes, 
but  does  not  vanish  identically,  can  be  written  in  one,  and 
essentially  in  only  one,  way  as  the  (finite  or  infinite)  product  of 
its  prime  factors. 

Moreover,  the  existence  theorem  for  such  functions,  whose 
prime  factors  are  arbitrary,  holds  there.  Let  Gi,  Gi,  •  •  •  be  an 
infinite  set  of  prime  functions  subject  merely  to  the  condition 
that  at  no  point  of  finite  space  do  the  monogenic  analytic  con- 
figurations which  correspond  to  their  roots  have  a  cluster  point. 

*  Gronwall,  Thesis,  Upsala,  1898,  p.  7. 

t  This  theorem  is  due  to  Gronwall,  1.  c.  It  was  rediscovered  by  Hahn, 
Monatshefte,  16  (1905),  p.  29. 


158  THE   MADISON   COLLOQUIUM. 

Then  there  exists  an  integral  function  whose  roots  are  those  of 
Gi,  G2,  •  •  •  and  which  include  no  other  points.* 

This  theorem  suggests  the  question,  what  are  the  character- 
istic properties  of  a  monogenic  analytic  configuration  M,  that  its 
finite  points  may  be  identical  with  the  roots  of  an  integral  function, 

(1)  G(z^,   ■■■,Zn)    ? 

First,  as  regards  the  function  G,  it  is  clear  that  this  must  be 
irreducible,  or  a  power  of  an  irreducible  function. 

Next,  let  (ai,  •  •  • ,  a„)  be  any  point  of  finite  space  which  is  a 
cluster  point  of  points  of  M.  A  necessary  condition  that  M  be 
given  by  (1)  is  seen  to  be  that  G{ai,  ■  •  • ,  an)  =  0.  Hence  all 
the  points  of  M  that  lie  in  the  neighborhood  of  (ai,  •••,  a„), 
and  no  others,  will  be  given  by  the  vanishing  of  a  finite  number 
of  functions,  G^■(sl,  •  •  •,  Sn),  each  analytic  at  (ai,  •  •  •,  a„)  and 
vanishing  there,  and  each  irreducible  there. 

Conversely,  this  condition  is  sufficient.  More  precisely,  let 
M  be  a  monogenic  analytic  configuration  of  the  {n  —  l)st  grade 
(  =  (n  —  l)-ter  Stufe)  in  the  domain  of  the  n  variables  (zi,  •  •  •, 
Zn),  and  let  it  be  such  that,  if  (ai,  •  •  •,  «„)  be  any  finite  cluster 
point  of  points  of  M,  then  the  points  of  M  which  lie  in  the 
neighborhood  of  (ai,  •  •  • ,  a„)  are  given  by  a  finite  number  of 
equations,  Gk{zi,  •  •  • ,  Zn)  =  0,  where  each  of  these  functions  is 
analytic  in  the  point  (ai,  •  •  • ,  a„)  and  vanishes  there,  and  more- 
over is  irreducible  there.  Then  the  finite  points  of  M  are 
coincident  with  the  roots  of  an  irreducible  integral  function 

This  theorem  was  stated  and  proven  by  Hahnf  for  the  case 
n  =  2,  the  formulation  there  being  slightly  different.  Halm 
also  states  more  general  theorems,  the  four-dimensional  space 

*  Appell,  Ada  Math.,  2  (1883)  p.  71.  Biermann,  Sitzungsber.  der  Wiener 
Akad.,  89  (1884),  2.  Abteil.,  p.  2G6.  Biermann  also  considers  the  generalization 
of  Mittag-LefBer's  theorem  in  its  more  restricted  form  to  functions  of  several 
variables.  Certain  wider  forms  of  the  theorems  can  be  treated  in  the  same 
manner. 

tL.  c. 


FUNCTIONS  OF  SEVERAL  COMPLEX  VARIABLES.      159 

of  the  variables  (x,  y)  not  being  the  whole  finite  space,  but  an 
arbitrary  cylindrical  space.  His  proofs  are  based  on  Cousin's 
theorem,  concerning  which  we  have  reported  at  length,  and  his 
conclusions  are,  therefore,  restricted  in  the  same  measure  as 
Cousin's  theorem  is  restricted.  Thus  Gronwall's  example  would 
vitiate  some  of  Hahn's  theorems  in  the  generality  in  which  Hahn 
stated  them. 


LECTURE   III 

SINGULAR   POINTS  AND  ANALYTIC   CONTINUATION 

§  1.     Introduction 

The  simplest  singular  points  wliich  an  analytic  function  of  a 
single  complex  variable  can  have  are  poles,  isolated  essential 
singularities,  and  branch-points. 

A  function  of  several  complex  variables  cannot  have  an  isolated 
singularity,  if  we  except  the  trivial  case  of  a  removable  singularity, 
i.  e.,  a  singularity  such  that  the  function  becomes  anal>i;ic  at 
the  point  in  question  when  a  suitable  value  is  assigned  to  it 
there.* 

For  example,  the  function  of  the  single  variable  z, 

f(^)  =  i 

has  an  isolated  singularity  at  the  point  2=0. 

But  the  function  of  the  two  complex  variables  tv  =  u  -{-  vi, 

z  =  X  -\-  yi: 

F{w,  2)  =  I , 

has  a  whole  two-dimensional  manifold  of  singularities  in  the 
four-dimensional  space  of  these  variables,  namely,  the  points 
{it,  ^^  0,  0). 

It  is  a  theorem  due  to  Weierstrass  and  proven  by  Rungef 
that  to  an  arbitrary  continuum  T  of  the  complex  s-plane  there 
correspond  functions  of  z  which  are  analytic  at  every  point  of  T 
and  which  furthermore  cannot  be  continued  analytically  over 

*  This  result  can  be  obtained  directly  from  Cauchy's  integral  formula  or 
Laurent's  series.  It  was  stated  by  Hurwitz  in  his  Zurich  address,  Verh.  des 
1.  intern.  Malh.-Kongr esses,  1897,  p.  104. 

t  Ada,  6  (1884),  p.  229. 

160 


FUNCTIONS   OF   SEVERAL  C0:MPLEX   VARIABLES.  161 

a  single  boundary  point  of   T.     This  theorem  has,  moreover, 
recently  been  extended  to  the  most  general  Riemann's  surface.* 
It  is  clear  from  the  foregoing  ihat  such  a  theorem  cannot 
hold  for  functions  of  more  than  one  variable. 

§  2,      XON-ESSENTIAL  SINGULARITIES 

The  analogue  of  a  pole  of  a  function  of  a  single  variable  is  a 
point  (ai,  •  •  •,  Qn),  in  whose  neighborhood  the  function  can  be 
written  in  the  form 

U;  -f*  (.21,    ■  •  • ,  2„)  =  777-  —:, 

h\Zi,   •  •  -,  Zn) 

where  G  and  H  are  both  analytic  at  (ai,  •  •  •,  a„),  and 

G(ai,  •  ",  cin)  =  0,     H{ai,  •  •  • ,  an)  +0. 

Here,  F  becomes  infinite  for  all  methods  of  approach  to  the 
point,  just  as  in  the  case  n  =  \.  We  shall  denote  such  a  point 
as  a  yole,  or  as  a  non-essentially  singular  'point  of  the  first  kind.] 

But  even  a  rational  function  can  have  a  more  complicated 
singularity.  Suppose  that  G  and  H  are  polynomials  relatively 
prime  to  each  other,  both  vanishing  at  (ai,  •  •  •,  a„);  e.  g., 

w 
F{ic,z)=—,  {a„a-^)  =  (0,0). 

Here,  the  function  can  actually  take  on  any  arbitrarily  assigned 
value  in  a  point  of  an  arbitrarily  assigned  neighborhood  of  the 
singular  point  in  question. 

We  are  led,  then,  to  a  second  kind  of  singularity,  the  function 
still  being  of  the  form  (1),  but  H  vanishing  also  at  the  point  in 
question,  though  still  being  prime  to  G.  Such  a  point  is  called 
a  iion-essentially  singular  point  of  the  second  kind.  In  the 
neighborhood  of  such  a  point,  which  we  will  take  as  lying  at  the 

*  The  question  has  been  treated  by  Koebe,  Freundlich,  and  Osgood;  cf. 
Osgood,  Funktionentheorie,  v.  1,  2d  ed.,  1912,  p.  747. 
t  Weierstrass,  Werke,  2,  p.  156. 


162  THE   MADISON    COLLOQUIUM. 

origin,  (0,  •  •  •,  0),  the  function  can  in  general*  be  written  in  the 
form: 

(2)    F(zu  ■'■,zn)=    ^j^^^^j-i_^,,.^Bi  "^'^'  "  • '  "")' 

where  the  coefficients  A,  B  are  functions  of  (zi,  •  •  •,  2„_i),  each 
analytic  at  the  point  (0,  •  •  • ,  0)  and  vanishing  there,  the  two 
polynomials  in  Zn  being  prime  to  each  other;  and  where,  moreover, 
12  is  analytic  and  not  zero  at  the  origin. 

In  every  neighborhood  of  a  pole  there  are  other  poles,  their 
locus  being  the  (2?i  —  2) -dimensional  analytic  manifold  or 
manifolds 

G(zi,  •  •  •,  z„)  =  0. 

But  there  are  no  other  singularities  in  the  neighborhood  in 
question.  For  the  special  case  n  =  2  the  non-essential  singu- 
larities of  the  second  kind  are  isolated  points,  since  two  functions 
G{iv,  z),  H{w,  z)  which  are  prime  to  each  other,  like  two  poly- 
nomials having  this  property,  can  vanish  simultaneously  only 
in  isolated  points.  But  when  n  >  2,  there  will  be  a  whole 
(2?i  —  4) -dimensional  locus  of  singularities  of  the  second  kind, — 
this  locus  consisting  of  a  finite  number  of  analytic  configurations, 
each  of  the  dimension  in  question.  In  fact,  the  necessary  and 
sufficient  condition  that  the  numerator  and  the  denominator  of 
the  fraction  in  (2)  vanish  simultaneously  is  that  their  resultant 
vanish.  The  latter  is  analytic  in  zi,  •  •  •,  Zn-i  and  vanishes  at 
the  origin;  but  it  does  not  vanish  identically. 

As  regards  the  poles  which  lie  in  the  neighborhood  of  a  singu- 
larity of  the  second  kind,  they  are  situated  on  the  manifold,  or 
manifolds, 

GiZi,    ■  •  ',  Zn)    =   0, 

and  they  consist  of  the  totality  of  such  points  with  the  exception 
of  those  for  which  //  also  vanishes,  i.  e.,  the  singularities  of  the 
second  kind. 


*  In  any  case,  a  suitable  homogeneous  linear  transformation  of  zi,  •  •  •,  Zn 
will  yield  a  new  function  for  which  the  statement  is  true;  cf.  IV,  §  1.  The 
theorems  of  the  paragraph  just  cited  are  assumed  in  the  present  paragraph. 


FUNCTIONS   OF  SEVERAL  COMPLEX   VARIABLES.  163 

A  function  which  has  no  other  singularities  in  a  given  region 
or  in  the  neighborhood  of  a  given  point  than  non-essential  ones 
is  said  to  be  meromorphic  in  the  region  or  in  the  point. 

§  3.    Essential  Singularities 

An  analytic  function  of  a  single  complex  variable  z  may  have 
an  isolated  essential  singularity,  z  =  a,  of  either  one  of  two  kinds: 
(a)  the  function  may  be  analytic  throughout  the  complete 
neighborhood  of  the  point  a  except  at  the  point  itself,  and  there 
neither  remain  finite  nor  become  infinite;  (6)  the  function  may 
have  poles  that  cluster  about  the  point  a,  being  analytic  at  all 
other  points  of  the  neighborhood  distinct  from  a. 

It  follows  from  the  first  theorem  of  §  1  that  the  first  case  has 
nothing  corresponding  to  it  when  we  pass  to  functions  of  several 
complex  variables.  But  may  not  the  second  case  be  realized? 
May  not  a  function  of  several  variables,  f{zi,  •  ■  ■ ,  Zn),  be  analytic 
except  for  non-essential  singularities  throughout  the  whole 
neighborhood  of  a  point  (oi,  •••,  a„),  this  point  alone  being 
excepted?  Weierstrass  believed  apparently  that  it  can,  for  he 
stated  the  following  theorem.* 

To  an  arbitrary  continuum  in  the  2/i-dimensional  space  of  the 
variables  (zi,  •  •  • ,  2„)  there  correspond  functions  analytic  or 
having  at  most  non-essential  singularities,  but  having  in  every 
boundary  point  a  singularity  of  higher  order. 

This  theorem,  however,  is  false,  as  was  shown  by  E.  E.  Levif 
in  a  notable  paper  published  three  years  ago,  to  which  we  shall 
return  later,  §§  8,  9.  In  particular,  it  appears  that  an  isolated 
essential  singularity  is  impossible. 

§  4.     Removable  Singularities 
In  his  inaugural  dissertation  RiemannJ  stated  and  proved  the 
theorem  whose  practical  value  is  so  well  known,  namely,  that 

*  Journ.fur  Math.,  89  (1880),  p.  o=Werke,  2,  p.  129. 

^  Ann.  di.  mat.  (3),  17  (1910),  p.  61.  Levi's  paper  appeared  while  a  paper 
of  Hartogs,  Math.  Ann.,  70  (1911),  p.  217,  overlapping  to  some  extent  Levi's 
paper  and  showing  in  particular  the  impossibility  of  an  isolated  essential 
singularity,  was  in  press. 

t  Gottingen  Dissertation,  1851,  §  12,  =Werke,  p.  23. 


164  THE   MADISOX   COLLOQUIUM. 

if  a  function  f(z)  is  analytic  throughout  the  neighborhood  T  of 
a  point  z  =  a  with  the  possible  exception  of  this  point  itself,  and 
if  f{z)  remains  finite  in  T,  then  f{z)  approaches  a  limit  when  z 
approaches  a ;  and  if  the  function  is  defined  for  z  =  a  as  equal  to 
its  limiting  value  there,  then  it  is  analytic  in  this  point  also. 

This  theorem  admits  a  number  of  generalizations  or  extensions 
for  functions  of  several  variables.  The  most  obvious  one  was 
stated  and  proven  by  Kistler*  in  the  following  formulation.  Let 
^(21,  •  •  • ,  Zn)  be  analytic  throughout  a  region  T  consisting  of  the 
neighborhood  of  a  point  (oi,  •••,  a„)  with  the  exception  at 
most  of  the  points  of  a  i^lri  —  2)-dimensional  anahi;ic  manifold 
L;  and  let  the  function  remain  finite  in  T.  Then  the  function 
will  approach  a  limit  in  the  points  of  L  and  will  be  analytic 
there  if  suitably  defined  there. 

Similarly,  Riemann's  theorem,  that  a  function  f{z)  which  is 
analytic  in  a  region  S  except  along  a  simple  regular  curve  C, 
where  it  is  continuous,  is  also  analytic  in  the  points  of  C,  can  be 
generalized.  If /(21,  •••,  Zn)  be  analytic  in  a  2??-dimensional 
region  except  in  the  points  of  a  single  {2n  —  l)-dimensional 
analytic!  manifold  (5,  where  /  is  continuous,  then  /  is  anah'tic  in 
the  points  of  S  also.     This  theorem  is  not  mentioned  by  Kistler. 

A  second  generalization  was  given  by  Kistler,  and  is  as  follows. 
Let  f{zi,  ■  •  • ,  Zn)  be  analytic  throughout  the  neighborhood  of  a 
point  (ai,  •  •  •,  (In)  with  the  exception  at  most  of  the  points  of  a 
finite  number  of  analytic  manifolds,  each  of  which  is  at  most 
(271  —  4)-dimensional.  No  hypothesis,  however,  is  now  made 
regarding  the  function's  remaining  finite.  Such  a  function  will 
be  analytic  in  the  excepted  points  also,  if  properly  defined  there. 
For  71  =  2,  this  becomes  the  first  theorem  of  §  L 


*  Gottingen  dissertation,  Ueber  Funktionen  von  mehroren  komplexen  Ver- 
andcrliclien,  §  7,  Basel,  1905.  This  theorem,  Hke  the  original  theorem  of  Rie- 
mann's, is  exceedingly  serviceable  in  practice,  and  was  probably  used  before 
Kistler's  enunciation  and  proof  of  it.  An  important  special  case  was  familiar 
toWeierstrass;  I,  §  1,  end.  A  second  proof  is  contained  substantially  in  Har- 
togs's  paper.  Math.  Ann.,  70  (1911),  p.  217. 

t  More  general  manifolds  are  also  admissible. 


FUNCTIONS  OF  SEVERAL  COMPLEX  VARIABLES.      165 

A  special  case  of  this  theorem  was  famihar  to  Weierstrass, 
namely,  that  in  which  the  function/  can  be  written,  in  the  neigh- 
borhood of  the  point  in  question,  as  the  quotient  of  two  functions, 
each  anal3i;ic  and  vanishing  there;  cf.  I,  §  1,  and  IV,  §  1. 

This  latter  theorem  of  Kistler's  admits  an  extension.  The 
excepted  points  may  fill  a  (2n  —  3) -dimensional  manifold,  the 
latter  being  such  that,  if  we  set  Zk  =  Xk  +  iyk,  then  three  of  the 
2n  coordinates,  as  yn-i,  Xn,  yn,  can  be  expressed  as  single-valued 
or  finitely  multiple-valued  continuous  functions  of  the  remainder 
in  the  neighborhood  in  question.  And  cases  reducible  to  the 
latter  by  linear  transformation  of  the  complex  variables  are 
obviously  included. 

Related  to  these  theorems  more  or  less  closely  is  a  further 
theorem  stated  by  Kistler,*  but  not  proven  by  him.  From  the 
neighborhood  T  of  a  point  (oi,  •  •  • ,  Qn)  let  the  points  of  a  set  L 
be  excluded,  L  consisting  of  the  points  of  a  finite  number  of 
analytic  manifolds,  each  of  dimension  2n  —  4  or  lower;  and  let 
the  remainder  of  T  be  denoted  by  T'.  In  the  region  T' 
f(zi,  •  •  • ,  Zn)  shall  be  meromorphic.  Then  the  function  can  have 
in  the  points  of  L  no  higher  singularities  than  removable  and 
non-essential  ones. 

The  proof  of  this  theorem  was  later  given  by  Hartogs.f 

We  note,  however,  in  closing  this  paragraph  an  interesting 
application  which  Kistler  makes  of  the  latter  theorem  to  a  proof 
of  Jacobi's  theorem  of  inversion,  I,  §  2. 

§  5.    Analytic  Continuation  by  Means  of  Cauchy's 

Integral  Formula 

During  the  last  few  years  a  number  of  important  theorems  on 
analytic  continuation  have  been  discovered,  chiefly  through  the 

*  L.  c.  In  the  light  of  Gronwall's  recent  discovery  concerning  the  scope 
of  Cousin's  theorem,  Eastler's  proof  was  even  more  restricted  than  appeared 
at  the  time. 

t  Math.  Ann.,  70  (1911),  p.  217.  Kistler  appears  to  have  had  no  substantial 
reason  for  supposing  the  theorem  to  be  true,  for  his  proof  is  based  on  a  mis- 
understanding of  Cousin's  results,  II,  §  9.  The  chief  credit  for  the  theorem 
would  seem,  therefore,  to  be  due  to  Hartogs. 


166  THE   MADISON   COLLOQUIOI, 

researches  of  Hartogs   and  E.   E.  Levi.      We  begin  with  the 
former. 

Hartogs' s  Theorem*  Let  B,  B'  be  regular  regions  of  the 
a:-plane  and  2/-plane  respectively,  and  let  K  be  the  neighborhood 
of  an  interior  point  yo  of  B'.  Letf{x,  y)  be  a  function  vAth.  the  fol- 
lo'vs'ing  properties,  cf .  Fig.  1 : 

(a)  In  the  interior  of  the  four-dimensional  cylindrical  region 
{B,  K),  f{x,  y)  shall  be  analytic;  and,  moreover,  for  every  point 
y'  of  K,f{x,  y'),  regarded  as  a  function  of  x  alone,  shall  be  con- 
tinuous on  the  boundary  C  of  B. 

(b)  For  every  point  ^  of  C,  /(^,  y),  regarded  as  a  function  of  y 
alone,  shall  be  analytic  witliin  B'  and  continuous  on  the  boundary 
C  of  B'. 

(c)  In  that  part  of  the  boundary  of  {B,  B')  which  is  determined 
by  the  points  (^,  77)  where  ^  ranges  over  C  and  r}  over  C',f{^,  77) 
shall  be  a  continuous  function  of  (^,  77). 

Then  f{x,  y)  can  be  continued  analytically  throughout  the 
interior  of  the  entire  cylindrical  region  {B,  B'). 

The  proof  of  this  theorem  is  simple.  For  every  interior  point 
(a*,  7j)  of  {B,  K),f(x,  y)  can  be  represented  by  Cauchy's  integral 
formula : 

Again,  by  Cauchy's  integral  formula, 

,,,  ,     1  rmn), 

/(-*)  =  2^  ii^''"- 
Hence 

where  the  double  integral  is  extended  over  the  part  S  of  the 

*  Sitzungsber.  der  Munchener  Akad.,  36  (1906),  p.  223.  The  formulation 
here  given  is  slightly  different  from  that  of  Hartogs. 


y 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARL\BLES. 


167 


boundary  of  {B,  B')  consisting  of  the  closed  surface  described 
in  the  condition  (c)  of  the  theorem. 

This  double  integral,  however,  represents  a  function  which  is 
analytic  in  the  two  independent  variables  (.r,  y)  throughout  the 
whole  interior  of  (B,  B'),  and  which,  furthermore,  coincides 
with  the  given  function  throughout  the  interior  of  {B,  K). 
Hence  the  proposition  is  established. 


a'-PLANE 


2/-PLANE 


Fig.  1. 


It  is  interesting  to  notice  the  nature  of  the  hypotheses,  (a) 
imposes  a  condition  on  the  function  in  each  point  of  a  4-dimen- 
sional  region;  (6)  is  3-dimensional,  in  that  it  is  made  up  of  a 
1-dimensional  system  of  2-dimensional  hypotheses;  while  (c)  is 
2-dimensional. 

Again,  the  points  of  (a)  form  a  4-dimensional  piece  of  the  4- 
dimensional  cylindrical  region  {B,  B').  The  points  of  (6)  form 
one  or  more  pieces  of  the  3-dimensional  boundary  of  {B,  B'). 
This  latter  manifold,  it  will  be  remembered,  consists  of  but  a 


168 


THE   MADISON    COLLOQUIUM. 


single  piece.  Finally,  the  points  of  (c)  yield  one  or  more  2-di- 
mensional  pieces  of  the  3-dimensional  boundary  of  (B,  B')  just 
referred  to,  and  they  also  lie  in  the  points  of  ih). 

A  further  aid  toward  a  geometric  realization  of  the  hypotheses 
is  obtained  if  we  picture  the  cylindrical  region  {B,  B')  as  a  rect- 
angle in  the  plane  of  analytic  geometry.  Here,  as  in  the  use  of 
that  plane  in  the  study  of  plane  cvu'ves  w^hen  the  complex  points 
are  admitted  to  the  discussion,  we  have,  it  is  true,  only  a  two- 
dimensional  figure  for  a  four-dimensional  set  of  geometric  objects; 
and  we  have  to  work  by  analogy. 


1/ 


•^<^'^^^'t^^y■^■>ky!■^^^;4^■:^■;/■^^■'-^ 


n 


l\ 


tn 


X 


Fig.  2. 


Condition  (a)  is  now  seen  to  refer  to  the  points  of  a  narrow 
strip  that  courses  the  large  rectangle  Imno,  the  latter  representing 
the  region  (5,  B').  Conditions  (6)  and  (c)  have  to  do  merely 
with  points  of  the  boundary,  which  lie  in  the  sides  lo  and  mn. 
In  the  conclusion,  the  function  is  extended  over  an  enlarged 
region  dimensionally  coordinate  with  the  slender  strip  of  con- 
dition (a). 

The  extension  of  the  theorem  in  the  above  formulation  to  the 
case  of  w-variables  is  obvious. 

For  three  variables,  the  geometric  interpretation  last  considered 
leads  to  a  rectangular  parallelepiped,  coursed  by  a  slender  one 
with  parallel  faces,  and  the  further  conditions  of  the  theorem 
are  interpretable  in  terms  of  regions  and  curves  lying  in  the  faces 
of  the  large  parallelepiped. 

Another  form  of  the  hypotheses  of  the  theorem,  somewhat 
less  general,  but  more  compact,  consists  in  requiring  the  function 
j{x\,  •  •  • ,  a-„)  to  be  analytic 


FUNCTIONS   OF   SEVERAL    COMPLEX   VARL\BLES.  169 

(a)  in  every  point  (xi,  ■  •  -,  Xk,  ff/t+i,  •••,«„),  where  av,  i  =  1, 
'  • ' ,  k,  ranges  over  Bi,  and  aj,  j  —  k  -{-  1,  •  •  •,  r?,  is  a  fixed  point 
in  Bj'y 

(b)  in  every  point  (^i,  •  •  •,  ^k,  Xk+i,  •  •  •,  Xn),  where  ^i,  i  =  1, 

•  •  •,  k,  ranges  over  the  boundary  Ci  of  Bi  and  Xj,  j  =  ^•  +  1, 

•  • ',  n,  ranges  over  Bj. 

The  function  will  then  admit  analytic  continuation  throughout 
the  cylindrical  region  (Bi,  •  •  •,  jB„). 

In  the  foregoing  results  is  contained  the  remarkable  theorem 
that  a  function /(.I'l,  •  •  •,  .t„)  which  is  analytic  in  every  boundary 
point  of  a  cylindrical  region  {Bi,  •  •  • ,  Bn)  admits  analytic  con- 
tinuation throughout  the  whole  region.* 

This  theorem  holds  for  the  general  case  of  any  four-dimensional 
region,  whether  cylindrical  or  not.     Cf.  §  9. 

§  6.    Application  to  the  Distribution  of  Singularities 

From  the  main  theorem  of  the  last  paragraph  Hartogs  deduces 
the  following  theorem  relating  to  the  distribution  of  the  singu- 
larities of  an  analytic  function. 

Theorem.  Let  /(.r,  y)  be  analytic  in  the  points  (0,  y),  where 
0<\y\<  h,  and  let/  have  a  singular  point  at  the  origin,  (0,  0). 
Then,  to  each  point  x'  of  a  certain  region  B:  \  x\  <  p,  will  cor- 
respond at  least  one  point  y'  of  the  region  B':\y\<h,  such  that 
/(.r,  y)  has  a  singular  point  in  {x',  y'). 

Here,  again,  it  is  useful  to  picture  the  points  to  ourselves  in 
the  plane  of  analytic  geometry.  We  assume  the  function 
f{x,  y)  to  be  analytic  along  that  part  of  the  ^-axis  which  lies  in 
the  neighborhood  of  the  origin,  this  latter  point  alone  being 
excepted  and  the  function  being  in  fact  singular  there.  The 
conclusion  is  that  the  projections  on  the  .r-axis  of  the  singular 
points  of  the  function  which  lie  in  the  rectangle  {B,  B')  com- 
pletely cover  that  part  of  the  axis  which  lies  in  this  rectangle. 

We  must  not,  of  course,  think  of  the  singular  points  as  dividing 
the  part  of  the  region  {B,  B')  in  which  the  function  is  considered; 

*  Hartogs,  1.  c,  p.  231. 


170  THE   MADISON   COLLOQUIUM. 

ill  two.  In  this  respect,  the  geometric  analogy  in  the  plane  is 
defective. 

Levi  has  given  a  similar  theorem  for  the  case  that  f(x,  y)  is 
allowed  to  be  meromorphic  instead  of  being  restricted  to  being 
analytic;  cf.  §  8,  Lemma  2.  The  formulation  of  that  lemma 
affords  a  more  precise  statement  for  Hartogs's  theorem. 

Continuation.  Singular  Surfaces.  The  theorem  of  the  pre- 
ceding paragraph  dealt  with  functions  /(.r,  y)  which  have  at 
least  one  singular  point  {x,  y)  in  the  neighborhood  of  the  origin 
(0,  0)  corresponding  to  every  x  near  a*  =  0.  We  turn  now  to  a 
theorem  which  has  to  do  with  functions  which  have  a  two-di- 
mensional assemblage  of  singular  points  spread  out  over  a 
surface. 

Theorem.^     Let 

y  =  ip{x) 

be  a  single-valued  continuous  function  of  the  complex  variable 
X  defined  throughout  a  certain  neighborhood  of  the  point  x  =  0. 
Let  f(x,  y)  be  analytic  at  all  points  of  the  neighborhood  of  the 
origin,  (0,  0),  with  the  exception  of  the  points  {x,  (p{x)),  and  let 
these  be  singular  points  of  f(x,  y)  which  are  not  removable 
singularities.     Then  (p{x)  is  an  analytic  function  of  x. 

The  point  of  this  theorem  is  the  very  great  restriction  to  which 
the  singularities  of  an  anah-tic  function  of  several  complex 
variables  are  seen  to  be  subject.  When  we  consider  that  the 
singular  points  of  an  analytic  function  of  a  single  complex  variable 
are  as  arbitrary  as  the  boundary  of  a  2-dimensional  continuum, 
the  essential  change  in  the  situation  on  passing  to  functions  of 
several  variables  becomes  evident. 

To  this  subject  belongs  a  theorem  of  Levi's,  to  which  we  shall 
turn  in  §  10. 

In  the  same  paper,  Hartogs  has  generalized  this  latter  theorem 
so  that  it  applies  to  functions  of  any  number  of  variables,  and 
these  functions  may  be  multiple-valued. 

*  Hartogs,  Ada,  32  (1908),  p.  57.  The  proof  of  this  theorem  is  complex, 
and  is  based  on  a  scries  of  earlier  developments. 


FUNCTIONS   OF   SEVERAL  COMPLEX  VARIABLES. 


171 


§  7.     Generalizations  of  the  Theorem  of  §5 

Hartogs  has  given  substantially  the  following  generalization 
of  the  theorem  of  §  5. 

Theorem.  Let  B  and  B'  be  regular  regions  of  the  x-  and 
2/-planes  respectively.  Let/(.T,  y)  be  a  function  with  the  following 
properties. 

(a)  j{x,  y)  shall  be  analytic  in  every  point  {^,y),  where  ^ 
ranges  over  the  complete  boundary  of  B  and  y  ranges  over  the 
interior  and  boundary  of  B'. 

(6)  f(x,  y)  shall  be  analytic  in  every  point  {x,  \l/(x)),  where 
^(x)  is  analytic  in  every  interior  and  boundary  point  of  B,  and 
where,  moreover,  the  points  y  =  \}/(x)  lie  within  B'. 

(c)  B'  shall  be  simply  connected.  More  generally,  the  points 
of  B'  which  correspond  to  the  points  of  B  through  the  function 
y  =  \l/(x)  shall  be  capable  of  being  enclosed  in  a  simply  connected 
region  lying  in  B'. 

Then  the  function /(.r,  y)  can  be  continued  analytically  through- 
out the  whole  4-dimensional  region  (B,  B'). 

The  geometric  picture  of  the  conditions  by  means  of  figures  in 
the  {x,  ?/)-plane  is  here  a  distinct  aid.  The  hypotheses  relate  to  a 
narrow  4-dimensional  region  which  encloses  part,  but  not  all, 
of  the  boundary  of  {B,  B'),  and  which,  furthermore,  penetrates 


Fig.  3. 


into  the  interior  of  {B,  B').  They  are  suggestively  indicated  by 
the  accompanying  figure.  In  one  minor  respect  this  represen- 
tation is  defective,  since  the  part  of  the  boundary  of  (5,  B')  to 
which  the  hypotheses  relate  does  not  necessarily  consist  of  more 
than  a  single  piece. 
13 


172  THE   MADISON   COLLOQUIUM. 

The  theorem  admits  of  generaUzation  to  functions  of  any 
number  of  variables;  Hartogs,  1.  c. 

Levi*  has  given  a  similar  theorem  for  the  case  that  f(x,  y)  is 
meromorphic  in  {B,  K)  and  also  in  the  points  (^,  y),  where  ^ 
lies  on  C  and  y  in  B'.    f{x,  y)  will  then  be  meromorphic  in  {B,  B') . 

§  8.    Levi's  Memoir  of  1910 

We  come  now  to  one  of  the  most  important  contributions  of 
recent  years  to  the  theory  we  are  discussing, — E.  E.  Levi's 
memoir  of  1910.*  With  the  aid  of  two  lemmas,  each  admitting 
a  simple  proof,  Levi  establishes  a  fundamental  theorem,  from 
which  follows  with  ease  a  complete  treatment  of  a  number  of 
questions  in  our  theory  which  had  presented  themselves  during 
the  last  decade. 

Lemma  1.     Let  f{x,y)  be  analytic  in  the  cylindrical  domain 

B:         \x\  ^h,     k  ^\y\  ^  K; 
and  let/(a:,  y)  be  developed  in  that  domain  into  a  Laurent's  series: 

(1)  M  2/)  =  E  gn{x)y\ 

In  order  that  the  analytic  continuation  of  f{x,  y)  into  the 

region 

\x\  ^  h,     \y\  <  h 

may  have  there  no  other  singularities  than  non-essential  ones, 
it  is  necessary  and  sufficient  that  there  exist  a  system  of  /  +  1 
functions  Ai{x),  i'  =  0,  1,  •  •  •,  /,  —  where  I  may  be  any  positive 
integer,  —  such  that 

AQ{x)gn-i{x)  +  Ai{x)gn-i+i{x)  +  •  •  •  +  Ai{x)gn{x)  =  0, 

where  n  =  —  1,  —  2,  •••.  Moreover,  the  functions  ^4,(.u) 
are  all  analytic  in  the  circle  \x\  ^  h,  and  Ai,{x)  does  not  vanish 
there,  t 

*  Ann.  di  Mai.  (3),  17  (1910),  p.  61. 

t  This  latter  function  may,  without  loss  of  generality,  be  set  equal  to  unity. 


FUNCTIONS   OF   SEVEEAL   COMPLEX  VARIABLES.  173 

That  the  condition  is  sufficient  appears  at  once  on  multiplying 
the  equation  (1)  through  by  the  function 

Ao{x)7/  +  Ai{x)y'-'  +  •  •  •  +  Ai{x) 

and  writing  the  right-hand  side  as  a  new  Laurent  series  in  y. 
The  negative  powers  of  y  are  seen  to  disappear,  and  hence  the 
function  represented  by  the  series  is  analytic  in  the  domain 
\x\Sh,\y\SK. 

The  proof  that  the  condition  is  necessary,  though  longer,  is 
not  complex. 

Definition.  A  function  /(.ti,  •  •  • ,  .t„)  shall  be  said  to  be 
continued  ineromorphically  from  the  point  A  to  the  point  B  along 
a  simple  path  L  of  2n-dimensional  space  if  the  function  is  mero- 
morphic  at  A  and  if,  on  enclosing  L  in  a  slender  simply  connected* 
271-dimensional  tube,  the  function,  when  continued  analytically 
within  this  tube,  presents  no  other  than  non-essential  singularities 
there.  The  resulting  function  will  then  be  single-valued  in 
those  points  of  the  tube  in  which  it  is  defined. 

A  function  can  be  continued  meromorphically  throughout  a 
region  if  it  can  be  continued  meromorphically  along  every  simple 
path  lying  in  the  region. 

Lemma  2.  Let  f{x,  y)  be  meromorphic  throughout  a  certain 
4-dimensional  continuum  T  containing  the  points  (0,  y)  where 
0  <  \y\  S  K,  but  not  containing  the  origin,  (0,  0) ;  and  let  it  not 
be  possible  to  continue  /(.r,  y)  meromorphically  to  the  origin 
along  a  path  lying,  except  for  its  extremity  (0,  0),  in  T. 

Then  there  exists  a  circle  S:  \x\  <  h,  such  that  (a)  the  points 
(x,  y),  where  x  lies  in  S  and  |  ?/  j  =  iv,  lie  in  T,  and  thus  f{x,  y) 
is  meromorphic  in  each  of  them;  (6)  to  each  point  x'  of  »S  cor- 
responds a  point  (.t'j  y'),  y'  =  r'e^\  r'  <  K,  such  that  /(.r,  y) 
can  be  continued  meromorphically  along  the  path  x  =  x', 
y  =  re^*  (where  r  is  the  independent  variable  and  x',  6  are  con- 

*  By  a  simply  connected  region,  or  more  precisely  a  linearly  simply  con- 
nected region,  is  meant  one  such  that  a  simple  closed  curve  lying  in  it  can  be 
drawn  together  continuously  to  an  interior  point  of  the  region  without  meeting 
the  boundary. 


174  THE  MADISON   COLLOQUIUM. 

slants,  and  where,  moreover,  r^  <  r  ^  K),  into  the  neighborhood 
of  (x'f  y'),  but  not  to  this  point. 

It  will  be  observed  that  if  the  conditions  of  the  theorem  are  ful- 
filled for  a  given  K,  then  they  are  also  fulfilled  for  any  smaller  K. 

We  can  picture  the  loci  a;  =  a:'  =  a,  [  o;  |  <  A,  as  surfaces  which 
form  afield  (in  the  sense  in  which  this  word  is  used  in  the  Calculus 
of  Variations)  in  the  4-dimensional  neighborhood  of  the  origin. 
If  we  make  any  analytic  transformation  of  this  neighborhood,  of 
the  form 

u  =  ip{x,  y),        V  =  xpix,  y), 

where  (p  and  \l/  are  functions  of  the  complex  variables  (a;,  y) 
analytic  at  the  origin,  and  where  the  Jacobian  of  <p  and  ^  does 
not  vanish,  we  can  then  state  an  obvious  corollary  of  Lemma 
2  for  the  surfaces  in  the  {u,  2j)-space,  into  which  the  surfaces 
X  =  a  have  been  carried. 

This  is  all  the  preparation  Levi  needs  for  his  main  theorem, 
to  which  we  now  turn. 

Theorem.  Let  £  be  a  perfect  set  of  points  in  the  4-dimensional 
space  of  the  complex  variables  x,  y;  and  let  0  be  a  fixed  point 
of  this  space.  Let  r  be  the  distance  from  0  to  a  variable  point 
of  E.  If  there  be  a  point  P  oi  E  for  which  r  has  a  relative  maxi- 
mum,* then  there  cannot  exist  a  function  f{x,  y)  which  is  mero- 
morphic  in  the  neighborhood  of  P  except  for  the  points  of  E, 
and  in  each  of  those  points  has  an  essential  singularity. 

More  precisely  stated,  the  conclusion  is  this.  Consider  the 
continuum,  T,  exterior  to  the  hypersphere  through  P  with  0 
as  centre  and  interior  to  a  small  hypersphere  with  P  as  centre. 
Then  there  cannot  exist  a  function  meromorphic  in  T  and  not 
admitting  meromorphic  continuation  at  P. 

Since  Hartogsf  has  proven  Lemma  2  for  the  case  that  f{x,  y) 
is  required  to  be  analytic  instead  of  being  allowed  to  be  mero- 

*  Maximum  is  here  to  be  understood  as  meaning  that  r  shall  not,  in  the 
neighborhood  of  P,  take  a  larger  value  than  at  P;  but  it  may  attain  that 
value  at  other  points  of  the  neighborhood. 

t  §  6,  first  theorem. 


FUNCTIONS  OF  SEVEKAL  COMPLEX  VARIABLES.      175 

morphic,  it  is  possible  to  enunciate  the  foregoing  theorem  for  the 
case  that  the  word  meromorphic  is  changed  throughout  to 
analytic,  and  moreover  essential  singularity  is  replaced  by 
singularity. 

Both  the  second  lemma  and  the  main  theorem  are  formulated 
here  more  generally  than  in  Levi's  paper.  Levi's  proof  applies, 
however,  to  the  extended  theorems. 

§  9.    Continuation.    Lacunary  Spaces 

Regular  Regions.  We  will  understand  by  a  regular  region 
of  m-dimensional  space  a  finite  m-dimensional  continuum, 
together  with  its  boundary;  the  latter  consisting  of  a  finite 
number  of  simple,  regular,  closed,  non-intersecting,  {m  —  1)- 
dimensional  manifolds.  It  is  obvious  that  this  definition  can 
be  formulated  more  generally,  but  the  above  is  sufficient  for 
our  present  purposes. 

Theorem  1.  Let  f{x,  y)  be  analytic  at  every  point  of  the 
boundary  of  a  regular  region,  T,  of  the  4-dimensional  space  of 
{x,  y).  Then/(.'K,  y)  admits  an  analytic  continuation  throughout 
T,  and  the  resulting  function  will  be  analytic  *  in  T. 

This  theorem  corresponds  to  Levi's  Corollary  I,  1.  c,  p.  11, 
but  is  more  general.  His  statement  of  his  corollary  is  defective. 
We  will  speak  of  the  proof  after  taking  up  the  proof  of  the  next 
theorem. 

In  particular,  then,  it  follows  from  the  foregoing  theorem 
that  an  analytic  function  of  two  complex  variables  cannot  have 
a  finite  lacunary  region,  around  which  the  function  is  analytic. 
Thus,  for  example,  no  function  /(.r,  y)  exists  which  is  analytic 
in  the  spherical  shell  bounded  by  the  hyperspheres  with  centres 
at  the  origin  and  of  radii,  r  =  1  and  r  =  1  +  5,  5  >  0,  and  which 
has  in  the  former  hypersphere  a  natural  boundary. 

As  has  already  been  pointed  out,  the  theorem  was  stated  and 
proven  for  cylindrical  regions  by  Hartogs. 

Theorem  2.     This  theorem  differs  from  Theorem  1  solely  in 

*  Cf.  II,  §  1. 


176  THE   MADISON   COLLOQUIUM. 

having   the   word   analytic   replaced   throughout   by   the   word 
meromorphic. 

The  second  theorem  can  be  proven  as  follows.  We  may 
without  loss  of  generality  assume  that  the  boundary  of  T  is 
pierced  by  an  arbitrary  ray  from  the  origin  at  most  in  a  finite 
number  of  points.  On  each  ray  which  enters  T  there  will,  then, 
be  a  finite  number  of  segments  lying  in  T.  Let  AB  be  such  a 
segment,  and  let  B  be  the  extremity  more  remote  from  the 
origin.  Continue /(x,  ?/)  meromorphically  from  i?  toward  ^.  If 
it  is  possible  to  reach  A  on  every  segment,  the  theorem  is  granted. 
If  not,  let  Q  be  the  first  point  on  AB  that  cannot  be  reached 
from  B. 

Thus,  when  all  segments  are  considered,  a  set  of  points  Q 
lying  in  the  finite  region  T  are  obtained,  and  this  set  is,  from  its 
source,  necessarily  closed.  I^et  P  be  one  of  its  points  whose 
distance  from  0  is  a  maximum.  Then,  in  that  part  of  the 
neighborhood  of  P  which  lies  outside  of  the  hypersphere  through 
P  with  its  centre  at  0,  f{x,  y)  is  meromorphic.  The  function 
must,  therefore,  by  Levi's  theorem,  §  8,  admit  a  meromorphic 
continuation  at  P,  and  here  is  a  contradiction. 

The  first  theorem  can  be  proven  in  a  similar  manner  by  the 
aid  of  Levi's  theorem  of  §  8,  stated  for  functions  required  to  be 
analytic  instead  of  being  allowed  to  be  meromorphic. 

It  thus  appears  that  an  analytic  function  of  two  complex 
variables  cannot  have  a  finite  lacunary  space  around  which  the 
function  is  meromorphic. 

This  latter  result  is  in  direct  contradiction  to  Weierstrass's 
theorem  of  §  3,  and  appears  to  be  the  earliest  proof  that  that 
theorem  is  false.  From  Lemma  2  it  follows,  however,  immedi- 
ately that  an  isolated  essential  singularity  is  impossible,  and 
thus  a  more  elementary  proof  is  afforded  of  the  incorrectness  of 
that  theorem. 


functions  of  several  complex  vael^bles.  177 

§  10.     Concerning  the  Boundary  of  the  Domain  of 
Definition  of  /(.r,  y) 

Let  2  be  a  simple  regular  3-dimensional  manifold  of  4-dimen- 

sional  space.     Then  2  can  be  represented  analytically  by  the 

equation 

<p{^\,  X2,  2/1,  yx)  =  0, 

where  x  =  Xi -\-  Lv-2,  y=l/i  +  W-2;  where,  furthermore,  <f  is 
continuous  together  with  its  first  partial  derivatives;  and  where, 
finally,  not  all  of  these  four  derivatives  vanish  simultaneously. 
We  will  restrict  ourselves  to  such  manifolds  2  as  correspond  to 
functions  (p  having  continuous  second  derivatives. 

Levi  raises  the  question:  Can  a  given  manifold  of  the  above 
description,  or  a  restricted  piece  of  it,  serve  as  part  of  the  boundary 
of  a  region  in  which  a  function  f{x,  y)  is  meromorphic,  but 
beyond  which  f{x,  y)  cannot  be  continued  meromorphically 
across  any  part  of  2?  In  other  words,  can  2,  or  a  piece  of  2, 
be  a  natural  boundary? 

He  finds  that  the  answer  is,  in  general,  negative;  since,  for 
it  to  turn  out  affirmative,  c?  must  satisfy  the  following  necessary 
condition.  Let  c  >  0  on  the  side  of  2  where /(.r,  y)  is  to  be  mero- 
morpliic.     Denote  by  (S(c?)  the  following  expression: 

(do  dw 
^?i7  + 

K^^i^yx      ^x.^yi)         K^x^^Vi      ^x^cy^)\^Xydy.      dx.dyj' 


where 


^-'=(^.)"+(iy' 


and  where  A/V,  Ao'V  denote  similar  expressions  in  yi,  y-2.  Then 
must  (^{(p)  ^0  in  all  points  of  2. 

If  ^  <  0  on  the  side  of  2  where  f{x,  y)  is  meromorphic,  then 
must  S(^)  ^0  in  all  points  of  2. 

From  this  result  it  follows  that  if  there  is  to  exist  a  function 


178  THE  MADISON  COLLOQUIUM. 

f\{x,  y)  meromorphic  on  one  side  of  S  and  having  S  as  a  natural 
boundary;  and  also  a  function /2(x,  y)  meromorphic  on  the  other 
side  of  2  and  also  having  S  as  a  natural  boundary,  then  must 

^i<p)  =  0. 

How  far  are  these  conditions  sufficient?  In  the  present  memoir 
Levi  shows  that  this  last  condition  is  sufficient;  namely:  If 
Q.((p)  =  0  in  every  point  of  S:  ^  =  0,  then  there  are  functions 
analytic  on  each  side  of  2,  but  having  2  as  a  natural  boundary; — 
all  this,  at  least,  when  2  is  suitably  restricted  in  extent. 

In  a  later  paper  Levi*  obtains  the  further  result,  that  if 
^(<p)  <  0  in  all  points  of  2 :  ^  =  0,  then  there  exists  a  function 
f{x,  y)  analytic  on  the  side  of  2  where  ^  >  0  and  having  2  as  a 
natural  boundary; — all  this,  at  least,  when  2  is  suitably  re- 
stricted in  extent. 

§  IL    A  Theorem  Relating  to  Characteristic  Surfaces 

An  analytic  surface  in  space  of  four  dimensions  may  be  repre- 
sented by  a  pair  of  equations: 

(1)  w(.Ti,  a-2, 2/1, 2/2)  =  0,     v{xi,  X2, 2/1,  2/2)  =  0, 

where  u  and  v  are  real  functions  of  the  four  real  variables,  analytic 
at  the  point  in  question,  their  Jacobian  with  respect  to  two  of 
the  variables, — say  2/1, 2/2> — not  vanishing  there. 

Levi-Civitaf  raises  the  following  question.  Suppose  two  real 
functions,  p  and  q,  are  given  along  such  a  surface,  and  are  analytic 
there.  Thus  p  and  q  may  be  any  functions  of  .ri,  X2  analytic 
at  the  point  in  question,  if  these  are  the  preferred  variables. 
Does  a  function  of  the  complex  variables  exist: 

iv{x,  y),    X  =  x-i.  +  ix2,    y  =  yi  +  iy^, 


*  Ann.  di  Mat.  (3),  18  (1911),  p.  69. 

t  Rendiconli  Accad.  Lincei  (5),  14  (1905),  p.  492.  He  prefaces  his  problem 
by  recalling  the  Cauchy  problem  for  two  independent  variables,  x  and  y,  and 
an  analytic  cur\'e  C  in  their  plane;  an  arbitrary  sequence  of  analytic  values 
being  assumed  along  C. 


FUNCTIONS  OF  SEVERAL  COMPLEX  VAKL\BLES.      179 

analytic  at  the  point  in  question  and  taking  on  the  value  p  +  qi 
along  the  surface  in  question, — all  this,  at  least,  in  a  certain 
neighborhood  of  the  given  point? 

He  finds  the  answer  to  be  affirmative  and  the  function  ic  to 
be  uniquely  determined,  provided  the  surface  is  not  what  he 
calls  a  characteristic  surface,  i.  e.,  a  surface  along  which  an  analytic 
function  of  two  complex  variables,  which  is  not  identically  zero, 
vanishes.  In  the  case  of  a  characteristic  surface,  there  will  in 
general  be  no  solution  of  the  problem.  Suppose,  for  example, 
that  the  surface  is  ?/  =  0, — and  the  general  case  of  a  characteristic 
surface  is  reducible  to  this  case.     Then 

w{x,  0)  =  p  +  qi, 

and  it  is  evident  that  f  -\-  qi  must  be  a  function  of  x  analytic 
at  the  given  point. 

If  this  condition  is  satisfied,  there  will  be,  not  a  single,  but  an 
infinite  number  of  solutions. 

From  these  results  follow  at  once  the  theorems: 

If  /(2*>  y)  is  anal}"tic  at  a  point  and  vanishes  along  a  non- 
characteristic  siu-face  through  that  point,  no  matter  how  re- 
stricted that  surface  may  be,  it  vanishes  identically. 

If  f(x,  y)  and  <p(x,  y)  are  both  analytic  at  a  point  and  take  on 
the  same  values  along  a  non-characteristic  surface  through  that 
point,  however  restricted  that  surface  may  be,  they  are  identi- 
cally equal  to  each  other. 

Le\d-Civita  extends  the  foregoing  theorems  to  functions  of  any 
number  of  variables. 

There  is  a  theorem  of  Levi's*  bearing  on  these  characteristic 
surfaces.  He  shows  that  any  three-dimensional  manifold 
ip  =  0  (§  10),  in  every  point  of  wliich  S(^)  =  0,  is  composed  of  a 
one-parameter  family  of  characteristic  surfaces. 

The  theorems  of  these  last  two  lectures  have  brought  out  clearly 
the  fact  that  the  analytic  functions  of  several  complex  variables 


*  Ann.  di  Mat.  (3).  17  (1910),  p.  89. 


180  THE   MADISON   COLLOQUIUM. 

are  far  less  capable  of  adapting  themselves  to  a  preassigned  region 
of  definition  than  is  the  case  with  the  functions  of  a  single  variable. 
An  explanation  is,  very  likely,  to  be  found  in  the  following  fact, 
to  which  we  have  already  called  attention  (II,  §  1) .  The  real  part 
of  an  analytic  function  of  a  single  variable  has  to  satisfy  but  a 
single  linear  partial  differential  equation  (Laplace's  equation). 
In  the  case,  however,  of  an  analytic  function  of  several  varia- 
bles, the  real  part  has  to  satisfy  a  simultaneous  system  of  such 
equations. 


LECTURE  IV 

IMPLICIT  FUNCTIONS 

§  1.    Weierstrass's  Theorem  of  Factorization 

The  following  theorem  is  due  to  Weierstrass.* 

Theorem  of  Factorization.  Let  F(u;  Xi,  •  •  •,  .r„)  be  a  function 
of  the  n -\-  I  variables  u,  Xi,  •••,  Xn,  analytic  in  the  origin 
(0;  0,  •  •  •,  0)  and  vanishing  there.     Let 

(1)  F(v;0,  ••-,0)  ^0. 

Then,  throughout  a  certain  neighborhood  of  the  origin, 

T:        \n\<  h,     \x,:\  <  h' ,  Jc=  1,  ■■■,n, 

the  following  equation  holds: 

(2)  F(ir,  xu  ■  ■  -,  Xn)=[u"'+Ai2r-'-\ [-A„,mu;xi,  •  •  •,Xn)] 

where  Ai  is  analytic  in  xi,  •  •  • .  .t„  throughout  the  region  l^^l  <^' 
and  vanishes  at  the  origin,  and  Q  is  analytic  in  u,  Xi,  ■  •  ■ ,  Xn 
throughout  T  and  does  not  vanish  there. 

If /(2o>  21,  •  •  •,  Zn)  is  any  function  of  Za,  z\,  •  •  •,  Zn,  analj-tic  at 
the  origin  and  vanishing  there,  but  not  vanishing  identically, 
it  is  possible  by  means  of  a  suitable  linear  transformation  of  the 
n  +  1  variables  Zq,  Zi,  •  •  • ,  Zn  to  carry  /  over  into  a  function 
Fill',  xi,  •  •  ■ ,  Xn)  satisfying  the  foregoing  conditions. 

Irreducible  Factors.  On  the  theorem  of  factorization  can  be 
based  a  theory  of  irreducible  factors  of  an  analytic  function 
analogous  to  the  theory  in  the  case  of  polynomials. f  First,  as 
regards  division.  If  F{zi,  •  -  • ,  Zn)  and  <E>(si,  •  •  • ,  Zn)  are  both 
analytic  in  the  point  (a)  =  (oi,  •  •  •,  o„)  and  <^  does  not  vanish 

*  Lithographed,  Berlin,  1879;  Funlctionenlelire,  1886,  p.  105  =  Werke  2, 
p.  135.  In  a  foot  note  of  the  page  last  cited  Weierstrass  says  that  he  has 
repeatedly  given  the  theorem  in  his  university  lectures,  beginning  with  1860. 

t  AVeierstrass,  1.  c. 

181 


182  THE  MADISON   COLLOQUIUM. 

identical  ly,  but  does  vanish  at  (a) ;  and  if,  in  the  neighorhood  of 
(a),  a  relation  of  the  form 

F(Zi,    ■■•,Zn)    =   Q(Zi,    •",  Z„)*(Zi,    •  •  •,  Z„) 

holds,  Q  being  analytic  at  (a),  then  F  is  said  to  be  divisible  by 
<E>  in  the  point  (a).  If  G(zi,  •••,  Zn)  is  analytic  in  the  point 
(a)  =  (ai,  •  •  • ,  ttn)  and  vanishes  there,  then  G  is  said  to  be  irre- 
ducible at  (a)  if  no  equation  of  the  form  exists: 

^(21,    •  •  •  ,  Zn)   =    Gi(Zi,    •  •  • ,  Zn)  GiiZi,    •  •  ',  Zn), 

where  Gi  and  G2  are  both  analytic  at  (a)  and  both  vanish  there. 

Two  irreducible  factors  are  equivalent  if  their  quotient,  taken 
either  way,  presents  at  most  removable  singularities. 

A  function  G{zi,  •  •  • ,  Zn)  analytic  at  (a)  and  vanishing  there,  but 
not  vanishing  identically,  can  be  written  in  one,  and  essentially 
in  only  one,  way  as  the  product  of  factors  each  irreducible  in  (a). 

A  factor  which  is  irreducible  at  a  given  point  is  not  necessarily 
irreducible  at  every  one  of  its  vanishing  points  which  lies  in  a  cer- 
tain neighborhood  of  the  point.  Hence  the  expression  of  a  func- 
tion at  a  given  point  as  a  product  of  factors  each  irreducible  at 
that  point  does  not  always  retain  this  character  when  that  point 
is  replaced  by  a  second  root  of  the  function  that  lies  in  the 
neighborhood  of  that  point. 

The  theorem  of  algebraic  geometry  that  two  curves  or  surfaces 
which  have  ever  so  short  an  arc  or  small  a  region  in  common, 
must  necessarily  have  a  whole  irreducible  piece  in  common, 
finds  its  counterpart  here.  Let  F{zi,  •  •  • ,  Zn)  and  $(zi,  -  •  -,  Zn) 
both  be  analytic  at  the  origin  and  vanish  there,  and  let  $  be 
irreducible  there.  If  F  vanishes  at  all  points  in  the  neighborhood 
of  the  origin  at  which  $  vanishes,  then  F  is  divisible  by  $. 

The  Roots  of  an  Analytic  Function  of  Several  Variables.  In 
the  case  of  analytic  functions  of  a  single  variable  the  roots  are 
isolated.  This  theorem  appears  to  be  lost  for  functions  of  several 
variables,  since  such  a  function  which  vanishes  at  all  has  an 
infinite  number  of  roots  clustering  about  any  given  root.  The 
theorem  admits,  nevertheless,  a  perfectly  good  generalization. 


FUNCTIONS   OF    SEVERAL    COMPLEX   VARIABLES.  183 

It  is  not  the  Individual  root,  but  the  monogenic  analytic  con- 
figurations which  are  made  up  of  the  roots  and  which  exhaust 
the  latter,  that  are  the  analogue  of  the  roots  of  a  function  of  a 
single  variable.  And  now  it  is  seen  from  the  factor  theorem 
that  the  number  of  such  configurations  which  course  the  neigh- 
borhood of  a  given  root  is  finite. 

Earlier  Sources,  As  appears  from  the  applications  already 
considered,  there  are  two  wholly  distinct  classes  of  theorems  at 
issue.  The  theorem  of  factorization  asserts  the  existence  of  an 
identity  in  7i  +  1  independent  complex  variables,  the  left-hand 
side  being  a  function  F(u;  Xi,  •  •  • ,  .Tn)  vanishing  at  the  origin, 
but  such  that  F{u;  0,  •  •  •,  0)  ^  0;  and  the  right-hand  side  being 
the  product  of  the  two  factors  described  in  detail  in  the  state- 
ment of  the  theorem.  This  theorem  is  universally  admitted  to 
be  due  to  Weierstrass. 

On  the  other  hand,  such  a  function  put  equal  to  0 : 

F{u;  xi,  ---yXn)  =  0, 

defines  an  implicit  function  of  n  arguments.  That  the  latter 
function  is  given  as  the  root  of  a  polynomial : 

where  the  ^^'s  are  all  analytic  in  a;i,  •  •  •,  Xn  at  the  point  in  ques- 
tion and  vanish  there,  follows,  it  is  true,  from  Weierstrass's  theo- 
rem. But  Weierstrass  was  not  the  sole  discoverer  of  this  theo- 
rem. The  theorem  is  contained  substantially  in  Cauchy's  Turin 
memoir  of  1831.*  In  that  paper,  Cauchy  showed  that,  to  each 
point  {xi,  •  ■  • ,  Xn)  lying  in  a  certain  neighborhood  of  the  point 
(ai,   ••■,  a„)  in  question,  correspond  precisely  vi  roots  of  the 

equation 

F{u;  xi,  •  ",  Xn)  =  0. 

Furthermore,  if  ^(u)  be  any  function  of  u  analytic  at  the  point 
M  =  0  and  vanishing  there,  and  if  the  above  m  roots  be  denoted 
by  III,  •  •  • ,  Um,  then  the  symmetric  function 

$(Wl)  +    •  •  •    +  ^(«m) 

*  Cf.  Exercices  d'analyse,  2  (1841),  p.  65. 


184  THE  MADISON   COLLOQUIUM, 

is  expressed  by  a  definite  integral  which'  is  seen  to  represent  a 

function  of  Xi,  -  •• ,  .t„  analytic  at  the  point  (oi,   •  •  •,  a„)  and 

vanishing  there.* 

In  order,  then,  to  obtain  the  implicit  function  theorem  it 

remains  merely  to  set 

$(w)  =  u^,  k  =  1,  ' '  - ,  m, 

and  then  express  the  elementary  symmetric  functions  by  the 
familiar  formulas  in  terms  of  the  Newtonian  sums, 

Furthermore,  Cauchy  applied  his  method  to  the  solution  of  a 
problem  in  implicit  functions,  namely,  to  the  development  of  a 
function  into  a  series  of  Lagrange.  Thus  this  noted  series,  so 
prominent  in  the  early  history  of  the  theory  of  functions,  again 
makes  contact  with  modern  analysis. 

There  are  two  other  proofs  of  the  implicit  function  theorem 
considered  above,  both  of  which  antedate  Weierstrass's  publica- 
gion  in  the  Funktionenlehre,  namely,  Poincare's  and  Neumann's,  f 

§  2.    A  Tentative  Generalization  of  the  Theorem  of 

Factorization 

In  the  case  n  =  1,  in  which  F  depends  on  only  two  variables, 
u,  X,  it  is  possible  to  dispense  with  the  condition  (1)  altogether, 
provided  F{ii,  x)  does  not  vanish  identically,  the  relation  (2) 
being  then  modified  as  follows: 

F{u,  x)  =  x^  {iC^  +  .4i?r-'  +  •  •  •  +  ^„0  Q.{u,  x), 

where  Z  is  a  positive  integer,  or  0.     Even  the  proviso  just  men- 
tioned can  be  avoided  if  we  write 

F{u,  x)  =  {Aou^  +  Aiv^-'  +  .  .  .  +  A„,)  U(n,  x). 


*  The  work  is  carried  through  for  the  case  n  =  1,  our  function  F{u;  Xi,  •  •  • ,  x„) 
being  represented  there  by/(a-,  y)  and  the  above  function  ^(u)  by  F{y). 

Th(!  proof  of  the  theorem  of  factorization  given  by  Goursat,  Cours  d'analyse, 
2,  §  35G,  is  based  on  Cauchy's  analysis. 

t  Poincar6,  Paris,  Th6se,  1879,  pp.  6,  7.  Neumann,  Leipziger  Berichle,  35 
(1883),  p.  85;  Abelsche  Integrale,  2d  cd.,  1884,  p.  125. 


^v     FUNCTIONS   OF   SEVEEAL   COMPLEX  VAEIABLES.  185 

Thus  we  have  a  form  of  the  factor  theorem  which  holds  in  all 
cases  and  which  does  not  depend  on  an  eventual  change  of  the 
independent  variables  by  a  linear  transformation. 

A  corresponding  form  for  the  general  case,  n  >  1,  would  be  a 
valuable  contribution,  since  it  is  not  always  feasible,  under  the 
conditions  of  the  problem  in  hand,  to  make  the  above  linear 
transformation.     The  tentative  theorem  is  as  follows. 

Tentative  Theorem.  Let  F{u;  xi,  •••,  Xn)  be  analytic  at  the 
origin  and  vanish  there.  Then,  throughout  a  certain  neighbor- 
hood of  the  origin, 

T\         \u\  <  h,     \xk\  <  h',  k  =  1,  •  •  •,  n, 

the  following  equation  holds: 

F(u;  xi,  •••,  Xn)=(Aou"'+Aiu'^-^-\ f-^m)^(w;  xi,  ■■-,  Xn), 

where  Ak,  k  =  0,  I,  ■  •  • ,  n,  is  analytic  in  X\,  •  " ,  Xn  throughout 
the  region  |  xi  \  <  h'  and  vanishes  at  the  origin  when  Z:  >  0;  and 
where  12  is  analytic  in  T  and  does  not  vanish  there.  Aq  may  or 
may  not  vanish. 

For  polynomials  the  theorem  is  obvious.  I  have  not  succeeded 
in  proving  it  in  the  general  case  except  when  n  =  1.  But  in  my 
attempts  at  a  proof  I  have  seen  nothing  that  discredits  the 
theorem  and  much  that  renders  it  probable.  I  tliink  the  chances 
are  that  the  theorem  is  true,  and  I  hope  that  someone  wiU 
investigate  this  question. 

§  3.    Algebroid  Configurations 
Consider  the  function  defined  by  the  equation 

(1)  F=  u^  +  A,ii^~'  +  .  •  •  +  J„.  =  0, 

where  Ak{xi,  ■••,  .i*„)  is  analytic  in  the  point  {x)  =  (0)  and 
vanishes  there,  and  the  polynomial  is  irreducible.  Such  a 
function  is  called  an  algebroid  function* 

*  Poincare,  These,  1879,  p.  4.  It  is  sometimes  desirable  to  admit  the  case 
that  the  coefficient  of  u'"  is  a  function  ^o(xi,  •  •  •,  x„)  analytic  at  the  point 
(x)  =  (0)  and  vanishing  there. 


186  THE  MADISON   COLLOQUIUM. 

Let 

A(.Ti,    •  •  •,  Xn) 

be  the  discriminant  of  F.  Then  A  ^  0,  and  to  every  point  (x) 
in  the  neighborhood  of  (x)  =  (0)  in  which  A  4=  0  there  correspond 
m  distinct  roots  of  F.  These  may  be  so  grouped  as  to  yield  m 
functions  Ui,  •  •  • ,  «m,  each  analytic  in  a  preassigned  point  in 
which  A  #  0.  Moreover,  one  of  these  functions  can  be  con- 
tinued analytically  into  every  other  one,  and  thus  they  are  all 
elements  of  one  and  the  same  monogenic  analytic  function. 

If  71  =  1,  we  are  led  to  an  ordinary  Riemann's  surface  with  a 
single  branch  point  in  the  point  x  =  0,  in  which  all  m  leaves  hang 
together. 

If  w  >  1,  it  is  still  convenient  to  think  of  a  Riemannian  mani- 
fold $  of  m  sheets,  or  leaves,  as  we  will  still  say;  though  these 
leaves  are  no  longer  surfaces,  but  2n-dimensional  manifolds. 

We  meet  here,  however,  an  entirely  new  order  of  relations. 
In  the  case  n  =  1,  there  was  but  a  single  branch  point.  That 
was  fixed,  and  the  junction  lines  were  movable  and  to  a  large 
extent  arbitrary.     Here,  however,  the  whole  locus 

(2)  A(.Ti,    -",Xn)   =   0 

yields  points  for  which  two  or  more  of  the  w^■'s  coincide.  In 
such  a  point,  two  u's  which  coincide  may  or  may  not  belong  to 
functions  each  analytic  at  the  point  in  question  and  satisfying 
the  equation  F  =  0.  In  the  former  case,  the  Riemann  manifold 
$  has  a  multiple  (2/i  —  2)-dimensional  manifold,  like  a  multiple 
point  of  a  plane  curve  at  which  all  the  tangents  are  distinct  and 
non-vertical,  or  more  generally,  at  which  no  two  branches  are 
connected  with  each  other. 

In  the  latter  case,  however,  we  have  a  whole  {2n  —  2)- 
dimensional  manifold  of  branch  points,  and  the  corresponding  w's 
are  not  analytic  in  (x)  at  such  a  point.  In  other  leaves  above  or 
below  such  a  point  it  may,  of  course,  happen  that  the  corre- 
sponding determinations  of  u  are  analytic. 

There   still   remain,   in   addition,   the  junctions.     These  are 


FUNCTIONS   OF    SEVERAL   COMPLEX  VAMABLES.  187 

(2n  —  l)-dimensional  manifolds,  largely  arbitrary  in  location 
and  character,  but  necessarily  passing  through  the  loci  of  branch 
points,  i.  e.,  the  branch  manifolds,  and  along  these  junctions 
one  branch  of  the  function  goes  over  into  another  branch,  re- 
maining analytic  all  the  while. 

A  simple  example  or  two  will  serve  to  illumine  the  above 
relations. 

Example  1. — 

u^  —  X  =  0, 

the  independent  variables  being  two  in  number,  x  and  y.  Here, 
the  space  of  the  independent  variables  is  a  four-dimensional 
real  space  Ra,  corresponding  to  the  tw^o  spheres, — the  .x-sphere 
and  the  ^/-sphere.  If  we  set  x  =  .ti  +  1x2,  y  =  y\-\-  iy^,  the 
points  of  i?4  wdll  be  (.x'l,  .T2,  y\,  2/2).  The  Riemann  manifold  $ 
is  two-leaved.     The  branch  manifold  consists  of  the  surfaces: 

S2':     (0,0,2/1,2/2);         S2":     (00,00,2/1,2/2), 

where  the  point  (2/1,  2/2)  ranges  over  the  whole  2/-sphere. 
As  the  junction  we  may  take  the  3-dimensional  manifold 

Rz'.         (-Ti,  0,2/1,  2/2),     0  ^  a:i  ^  CO, 

i.  e.,  the  point  (.I'l,  X2)  is  any  point  of  the  positive  axis  of  reals, 
including  the  points  x  =  0  and  .x  =  co  ;  and  (2/1,  2/2)  ranges  in- 
dependently over  the  extended  2/-plane.  There  is  wide  latitude 
in  the  choice  of  Rz,  but  it  must  contain  the  surfaces  S^  and  *S2". 

Example  2. — 

u~  —  xy"^  =  0, 

the  independent  variables  again  being  x  and  y. 

Here,  the  two  values  of  w  become  equal,  not  only  in  the  points 
of  the  above  surfaces  S2'  and  S2",  but  also  in  the  points 

®2':     (.Ti,  x.,  0,  0);         (B2":     (.ri,  X2,  o),  o)). 

Nevertheless,  in  the  neighborhood  of  any  point  of  ©2'  and  ©2" 
which  does  not  lie  on  ^2'  or  So"  the  values  of  u  can  be  grouped  so 
as  to  yield,  two  functions,  each  single-valued  and  analytic  through- 
out the  neighborhood  in  question. 

The  Riemann  manifold  $  may  be  taken  precisely  as  before. 
14 


188  the  madison  colloquium. 

§  4.     Continuation.    The  Branch  Points  of  the 

Discriminant 

It  is  important  to  notice  how  the  dependent  variable  behaves 
in  the  points  of  a  (2n  —  2) -dimensional  manifold  of  branch 
points.  If  we  are  at  liberty  to  make,  if  necessary,  a  non-singular 
linear  transformation  of  the  x's,  we  may  assume  that 

A(0,  ■■•,0,Xn)  +0, 

and  hence  replace  the  equation  A  =  0  by  an  algebroid  equation 

in  Xn.     Let 

(3)  D  =  xj  +  BixJ-'  +  ■■■  +  Bi  =  0, 

where  D  is  an  irreducible  factor  of  A;  and  let  Di{xu  ■  •  • ,  Xn-\) 
be  the  discriminant  of  D.  Then  Z)i  ^  0.  For  simplicity  in  the 
presentation,  we  confine  ourselves  to  the  case  that  A  has  no 
further  irreducible  factor. 

Consider  a  point  Pq:  (.ri°,  •  •  •  .t„_i°)  in  which  Di  4=  0.  In 
the  neighborhood  of  this  point  the  roots  of  (3)  can  be  grouped 
to  /  functions  Xn  ,  Xn" ,  •  •  • ,  Xn^^^  each  analytic  in  the  above  point 
and  all  elements  of  the  same  monogenic  analytic  function. 

If  we  substitute  one  of  these  elements,  xj,  in  the  coefficients 
of  (1),  the  new  polynomial, 

F=u"'+  Aiu"'-'  H h  Zn  =  0 

— where  Ak{xi,  •  •  -,  Xn-i)  is  analytic  in  the  point  (.ri°,  •  •  •,  Xn-i^) 
but  does  not  necessarily  vanish  there, — will  have  a  common 
factor  with  its  allied  polynomial 

-       dF 
F'  =  — . 

du 

Consider  the  greatest  common  divisor  of  F  and  F'.  Let  its 
irreducible  factors  be 

Gh{u,  Xi,  •  •-,  .r„_i),  A;  =  \,  •  •  -yV. 

In  general,  v  =  1  and  6'i  is  linear  in  u. 


FUNCTIONS    OF   SEVERAL   COMPLEX   VARIABLES.  189 

This  case  can  always  be  attained  by  a  linear  transformation : 

(4)  Xn  =   Xn  +  a  U, 

where  a  is  a  suitable  real  positive  number. 

In  fact,  returning  to  the  arbitrary  case  of  the  text,  let  Pi: 
{xi^,  •  •  • ,  Xn-i^)  be  a  point  of  the  neighborhood  of  Pq  in  which  the 
discriminant  of  no  Gk  vanishes ;  let  this  be  true  not  merely  for  the 
particular  element  Xn  that  was  substituted  in  F,  but  for  each  of 
the  other  I  —  1  determinations  of  .t„  given  by  (3),  Xn",  •  •  • ,  .t„^'\ 
Finally,  let  Pi  be  so  chosen  that  no  two  6r's,  —  whether  they 
belong  to  the  same  a*„^'^  or  to  different  ones:  .Tn^'^  and  Xn^^\  have 
equal  roots. 

Throughout  a  certain  neighborhood  of  Pi,  then,  we  have  I 
patches  2i,  •••,  S^  of  the  discriminant  manifold  (3),  and  the 
points  (a'l,  •  •  •,  Xn-i,  xj''^),  k  —  I,  •  •  • ,  I,  thus  defined  in  the 
space  of  (a'l,  •  •  • ,  Xn)  are  the  totality  of  the  points  of  so  much  of 
that  space  as  lies  in  a  certain  neighborhood  of  the  origin: 

|av|  <h,  j  =  1,  •  •  •,  n, 

for  which,  first,  {xi,  •  • ,  Xn-\)  lies  in  the  neighborhood  of  Pi  and, 
second,  (1)  has  multiple  roots.  Let  us  picture  the  corresponding 
points  («,  xi,  •  •  •,  Xn)  in  a  {2n  +  2)-dimensional  space,  where  u 
is  given  by  the  vanishing  of  the  different  Gk^ii,  Xi,  •  •  ■,  .r„_i), 
these  functions  being  taken  not  merely  for  .r,/,  but  also  for  the 
other  .I'n^^'^'s.  Then  these  (2?i  —  2)-dimensional  loci, — call  them 
®ij  •  •  •  J  ®f, — are  distinct  from  one  another.  They  are  analogous 
to  arcs  of  curves  lying  on  the  different  nappes  of  cylinders  whose 
directrices  are  the  /  2?z-dimensional  manifolds  defined  by  (3) : 

where  (.ri,  •  •  • ,  Xn-\)  lies  in  the  neighborhood  of  Pi. 

If  now  we  make  the  transformation  (4),  restricting  suitably 
the  positive  number  a,  then  the  configuration  P  =  0  will  go  over 
into  a  new  configuration  Pi  =  0  and  at  the  same  time  the  loci 
©1,  •  •  •,  ®t  will  go  over  into  loci  ®i',  •  •  •,  (S/  which  have  for 


190  THE   MADISON   COLLOQUIUM. 

the  transformed  configuration  Fi  =  0  the  same  meaning  that 
@i,  •  •  •,  @t  have  for  F  =  0,  i.  e.,  <Si,  •  •  •,  ©f  are  invariant  of 
the  transformation.  And  now,  if  the  neighborhood  of  Pi  is 
suitably  restricted,  the  number  of  the  ordinates  u  corresponding, 
for  a  given  (.ti,  •  •  •,  Xn-i),  to  the  ith  region  ®/,  i  =  1,  •  •  •,  f, 
will  reduce  to  unity.  Thus  the  new  I  will  equal  f ,  and  to  each  of 
the  new  points  {xi,  •  •  • ,  .t„_i,  Xn^*^)  will  correspond  but  a  single  u. 

It  thus  appears,  that,  in  general,  on  the  {2n  —  2) -dimensional 
analytic  manifold  or  manifolds  defined  by  the  equation  A  =  0 
the  multiple  roots  of  F  are  single-valued  and  analytic  except 
along  certain  {2n  —  4)-dimensional  analytic  manifolds. 

We  can  now  proceed  to  these  latter  manifolds  and  prove  a 
similar  theorem  for  them ;  and  so  on. 

The  reasoning  here  used  is  akin  to  that  employed  in  the  proof 
of  Weierstrass's  theorem,  §  8. 

§  5.    Single-Valued  Functions  on  an  Algebeoid 

Configuration 

Let  U  be  uniquely  defined  in  the  ordinary  points  of  the  alge- 
broid  configuration  (1),  i.  e.,  the  points  in  which  u  is  not  a  multiple 
root  of  (1),  and  let  it  be  analytic  in  such  points.  If,  furthermore, 
U  remains  finite,  then,  in  the  above  points, 

^  G{U,    Xi,    "',    Xn) 
F'{U,   Xi,    ••■,   Xn)' 

where  G  is  analytic  in  the  point  (0,  0,  •  •  • ,  0)  and  vanishes  when 
F'  vanishes  on  the  manifold.  Moreover,  U  is  an  algebroid  func- 
tion of  {xi,  •  •  • ,  a:„)  in  the  neighborhood  of  the  origin. 

It  would  be  a  mistake,  however,  to  think  that  when  U  satisfies 
the  above  hypothesis,  the  limiting  values  of  U  in  the  singular 
manifold  behave  as  did  the  coordinates,  u,  Xn,  etc.,  in  the  cases 
discussed  in  §  4.  The  following  example  will  show  what  may 
arise.     Let 

F=u-'-  x'iy'-  -  z'-){7/  -  kh^)  =  0, 

where  k  is  real  and  0  <  k  <  1,  and  the  independent  variables 


FUNCTIONS    OF   SEVERAL   COMPLEX   VARL\BLES.  191 

are  x,  y,  z.    Let 

X 

Then 

U-2  _    (^2  _   ^2)  (^2  _   ^.2,2)    ^    0. 

But  in  the  points  of  the  singular  manifold  x  =  0,  i.  e.,  in  the 
points  {u,  X,  y,  z)  =  (0,  0,  y,  z)  the  limiting  values  of  U  do  not 
form  a  function  single-valued  on  that  manifold. 

Connectivity  and  the  Riemann  Manifold.  In  the  preceding 
sections  we  have  taken  the  Riemann  manifold  <J>  for  granted. 
But  how  do  we  know  that  it  exists?  Even  for  functions  of  a 
single  complex  variable  this  question,  in  the  general  case,  was 
not  simple.  It  is  one  of  the  fundamental  problems  of  the  theory 
to  show  that,  to  any  monogenic  analytic  function  of  several 
complex  variables,  corresponds  a  Riemann  manifold.  One 
method  of  attack  would  be  to  prove  the  theorem  for  a  properly 
restricted  algebroid  configuration,  and  then  proceed  as  in  the 
case  of  functions  of  a  single  variable.* 

Consider  so  much  of  the  algebroid  configuration  (1),  §  3,  as 
lies  in  the  region  |  m  1  <  A',  |  .r^  |  <  lu.  The  linear  connectivity 
of  the  corresponding  Riemann  manifold  is  not  necessarily 
unity,  no  matter  how  far  k,  hi  be  restricted.  For,  in  particu- 
lar,  F  may  be  a  non-specialized  homogeneous  polynomial,  so 
that  (1)  is  the  equation  of  a  non-specialized  algebraic  cone  of 
degree  m. 

Parametric  Representation  im  Kleinen.  One  other  theorem  we 
will  mention, — the  theorem  of  the  parametric  representation  of 
an  analytic  configuration  im  Kleinen. 

Let  G{zi,  •  •  •,  Zn)  be  a  function  of  Si,  •  •  •,  Zn  analytic  at  the 
origin  and  vanishing  there,  but  not  vanishing  identically.  Then 
there  exists  a  finite  number  of  systems  of  equations 

(1)  Zk  =  <pk{ti,  •  ■  •,  tn-\),  k  =  1,  •  •  •,  n, 

where  ^^C^i,  •  •  •,  <n-i)  is  analytic  at  the  origin  (t)  =  (0)  and  van- 
*  Cf.  Osgood,  Lehrbuch  der  Funktionentheorie,  vol.  1,  2d.  ed.,  1912,  p.  747. 


192  THE   I\LU)ISON   COLLOQUIUM. 

ishes  there,  such  that,  throughout  a  certain  neighborhood  of  that 
point,  the  numbers  Zi,  •  •  ■ ,  Zn  thus  defined  are  roots  of  G.  And 
conversely,  to  each  root  of  G  within  a  certain  neighborhood  of 
the  origin  there  corresponds  at  least  one  system  (1)  which 
yields  this  point;  and  in  any  such  system  there  corresponds  but 
one  point  (t)  to  the  given  (z),  provided  (2)  ^  (0). 

For  the  case  n  =  2  the  proof  of  this  theorem  is  readily  given 
both  by  the  methods  of  Riemann  and  by  those  of  Noether  (de- 
veloped originally  for  algebraic  curves).  For  n  =*3,  after  an 
unsuccessful  attempt  by  Kobb,  a  proof  was  given  by  Black.* 
The  latter 's  methods  appear  to  suffice  for  the  general  case;  but 
detailed  algebraic  work  remains  to  be  done.  Weierstrass  states 
the  theorem  as  true  in  all  cases,  n  =  n.j 

§  6.     Solution  of  a  System  of  Analytic  Equations. 
Weierstrass's  Theorem 

There  is  a  second  theorem  of  Weierstrass's t  which  is  less  well 
known  than  the  factor  theorem,  but  which  deserves  a  prominent 
place  in  the  theory.     It  is  as  follows. 

Let 

(1)  Gi{zi,  ■■',Zn),     •••,     Gi{zi,  •■■,Zn),  l<  n, 

be  a  system  of  functions  each  analytic  at  the  origin  and  vanishing 
there,  but  not  vanishing  identically.  Then  the  roots  of  these 
functions,  regarded  as  simultaneous,  which  lie  in  the  neighborhood 
of  the  origin  can  be  represented  as  follows.  A  suitable  non- 
singular  linear  transformation  being  made: 

(2)  Zk  =   ttkl^Oi  +    •  •  •   +  O.knlCn,  k  =    1,    •••,11, 

the  values  of  iv  which  correspond  to  roots  of  the  original  functions 
(1)  will  be  given  by  a  finite  number  of  systems  of  equations  of 
the  following  type : 

*  Proceedings  Amer.  Acad,  of  Arts  and  Sci.,  37  (1902),  p.  281. 
t  Werke,  3,  pp.  103^. 
X  Werke,  3,  pp.  79-80. 


(3) 


FUXCTIOXS    OF    SEVERAL    COMPLEX   VAEL\BLES.  193 

'Fl(Wi,    •  ■  ■,  Wm,  ll'm+l)   =   «'m  +  l  +  Ai  W'^+Z  +    •  •  •   +  Ay  =  0, 


where  Ak  is  anal}i;ic  in  Wi,  -  •  • ,  Wm  at  the  origin  and.  vanishes 
there,  and  Fj  is  analytic  in  ivi,  •  •  • ,  Wm,  Wm+i  at  the  origin  and 
vanishes  there.  Fi{wi,  •  •  •,  Wm,  Wm+i)  is  irreducible  at  the  origin, 
and 


F[  = 


^U^m+l  ' 


To  each  root  (s)  =1=  (0)  of  (1)  lying  in  the  neighborhood  in 
question  corresponds  at  least  one  system  (3)  such  that  F'^  does 
not  vanish  in  the  point  (w)  which  is  the  image  of  (2).  Con- 
versely, each  system  of  values  (w)  lying  in  a  certain  neighbor- 
hood of  the  origin  and  satisfying  (3)  yields  a  root  (2)  of  (1)  lying 
in  the  neighborhood  of  (2)  =  (0). 

\Mien  the  conditions  of  the  problem  are  such  that  all  n  variables 
2i,  •  •  • ,  Zre  are  coordinate,  so  that  the  transformation  (2)  is 
available,  this  theorem  yields  complete  and  satisfactory  informa- 
tion regarding  the  solution  of  equations  (1)  im  Kleinen. 

The  proof  of  the  theorem  is  direct,  and  is  given  by  means  of 
the  factor  theorem  and  the  algorithm  of  the  greatest  common 
divisor. 

§  7,     CoxTixuATiox.     A  Gexeral  Theorem 

It  may  happen  that  the  variables  with  respect  to  which  it  is 
desired  to  solve  may  not  be  interchanged  with  the  remaining 
variables,  so  that  the  factor  theorem  is  not  available.  In  this 
case  the  following  theorem  may  be  useful.*  The  proof  is  closely 
allied  to  that  of  Weierstrass's  theorem,  §  6.  The  case  ?i  =  2  is 
covered  by  a  theorem  of  Bliss's,  f 

*  The  theorem  is  suggested  by  a  theorem  of  Poincare's,  These,  1879, 
Lemma  IV,  p.  14.  It  is  not  clear  what  Poincare  meant  bj'  the  words:  "...  si 
les  equations  <pi  =  <P2  =  •  •  •  =  <Pp  =  0  restent  distinctes  quand  on  annule 
tons  les  X.  .  .  ." 

t  Princeton  Colloquium,  1909,  published,  1913,  p.  71. 


194  THE   MADISON   COLLOQUIUM. 

Theorem.     Let  the  functions 

(1)  $fc(wi,  •  •  •,  2ii;  xi,  ■■',  Xp),  k  =  1,  •••,/, 

be  analytic  in  the  point  (u;  x)  =  (0;  0)  and  vanish  there.  Fur- 
thermore, let  the  I  equations 

(2)  ^k{uu  •  •  •,  wz;  0,  •  •  •,  0)  =  0,        k=  1,  "-,  I, 

admit  no  other  solution  than  (u)  =  (0)  in  the  neighborhood  of 
this  latter  point.  Then,  to  each  point  {x)  in  a  certain  neighbor- 
hood of  (x)  =  (0),  with  the  exception  of  those  which  lie  on  a 
locus  D  presently  to  be  considered,  there  will  correspond  N 
distinct  points,  {ui^^\  •••,  Ui^^^),  j  =  1,  •••,  N,  and  hence  N 
distinct  points  (ui^^\  •  •  • ,  ui^^^ ;  xi,  •  -  •  ,Xp),  such  that  the  equations 

(3)  ^k{ui,  •••,ui;  xi,  •■•,  Xp)  =  0 

are  satisfied  in  these  points.     N  is  a  fixed  positive  integer. 

Moreover,  these  are  the  only  points  of  the  neighborhood  of 
{u;  x)  =  (0;  0)  in  which  these  equations  are  satisfied  and  for 
which  (x)  does  not  lie  on  D. 

These  points  are  determined  as  follows,  ui  is  giVen  by  an 
algebroid  equation  having  no  multiple  factors, 

(4)  ui'' +  Aiui""-' +  ...  +^^v=0, 

where  Ak(xi/ -  •  • ,  Xp)  is  analytic  in  the  point  (.t)  =  (0)  and 
vanishes  there.  If  {x)  does  not  lie  on  D,  the  roots  of  (4)  are  all 
distinct,  and  analytic  in  {x) ,  and  the  further  functions  wi,  •  •  • , 
Ui-\  which  enter  to  form  the  roots  of  (1)  are  also  single-valued 
and  analytic  on  the  analytic  configuration,  or  configurations, 
(4)  except  at  most  for  points  for  which  {x)  lies  on  D. 

The  points  of  D  are  those  whose  co-ordinates  satisfy  at  least 
one  of  a  finite  number  of  equations 

I>\(xi,  •  •  •,  .Tp)  =  0,     •  •  •,     Dq{xi,  ■••,Xp)  =  0, 

where  Dk  is  analytic  in  the  point  (x)  =  (0)  and  vanishes  there, 
and  is  irreducible. 


FUNCTIONS   OF   SEVEKAL   COMPLEX   VARIABLES.  195 

We  can  go  further  and  say:  To  each  point  (.I'l,  •  •  •,  a-p_i)  of 
the  neighborhood  of  the  origin,  which  does  not  lie  on  a  certain 
exceptional  manifold  E,  there  correspond  M  distinct  points  of  D, 
and  likewise  M  distinct  points  of  the  neighborhood  of  (u;  x) 
=  (0;  0),  for  which  the  equations  (1)  are  satisfied. 

These  points  are  determined  as  follows.  Xp  is  given  by  an 
equation  having  no  multiple  factors : 

(5)  V+  Bix/'-'  +  . . .  +  5v  =  0, 

where  Bk(xi,  •  •  •,  Xp-y)  is  analytic  at  the  origin  and  vanishes  there. 
If  {x\,  "  -,  Xp-\)  does  not  lie  on  E,  the  roots  of  (5)  are  all  distinct 
and  analytic  in  {xi,  •  •  • ,  Xp-\)  and  the  further  functions  u\,  '  •  •  yUi 
which  enter  to  form  the  roots  of  (1)  are  also  single- valued  and 
analytic  on  the  analytic  configuration  or  configurations  (5) 
except,  at  most,  for  those  points  for  which  (a*i,  •••,  .ip-i)  lies 
on  E. 

The  points  of  E  are  those  whose  co-ordinates  satisfy  at  least 
one  of  a  finite  number  of  equations 

Ei{xi,  •  •  •,  a-p-i)  =  0,     •  •  •,    Ek{xi,  ■  • ',  .Tp_i)  =  0, 

where  Ek  is  analytic  in  the  point  {xi,  •  •  • ,  a'p_i)  =  (0,  •  •  • ,  0) 
and  vanishes  there,  and  is  irreducible. 

We  can  now  proceed  to  treat  the  points  of  E  in  a  similar  way, 
and  so  on. 

The  foregoing  formulation  is  deficient  in  one  respect.  In 
excepting,  as  the  first  step,  all  points  whose  (.r)  belongs  to  D 
some  points  w^ere  lost  which  have  not  later  been  regained. 
Consider  the  multiply  sheeted  Riemann  manifold  corresponding 
to  (4).  For  a  given  point  of  D  one  point  at  least  of  this  manifold 
is  to  be  excluded.  But  it  may  happen  that  points  above  or 
below  this  one,  in  other  sheets,  are  such  that  ui  and  the  other  li^'s 
will  be  analytic  there.  The  number  of  such  systems,  {iii,  ■  ■  ■  ,ui; 
i*^i>  *  •  •  >  ^p)  will,  however,  be  less  than  N. 

It  would  be  possible  to  give  to  this  theorem  a  formulation  more 
closely  resembling  that  of  Weierstrass's  theorem,  §  6. 


196  THE   MADISOX   COLLOQUIUM. 

Poincare  has  given  a  further  theorem,*  which  he  regards 
merely  as  another  form  of  the  theorem  of  his  Lemma  IV,  cited 
above.  Under  the  h\'potheses  of  the  last-named  theorem  he 
states  that  the  system  of  equations  (3)  can  be  replaced  by  an 
equivalent  sj'stem: 

^k{ui,  •  •  -,  ui;  xi,  •  •  •,  .Tp)  =  0,       h  =  I,  •  '  -,1, 

in  which  "^k,  in  addition  to  satisfying  the  conditions  imposed  on 
$/.-,  is  a  polynomial  in  Wi,  •  •  • ,  uu 

Special  Cases  of  the  Foregoing  Theorem.  A  special  case  of 
this  theorem  has  recently  been  investigated  by  MacMillan.f  It 
is  evident,  in  the  light  of  ^yeie^strass's  theorem,  that  no  one  of  the 
functions  ^i{ui,  -  •  -,  ui;  0,  •  •  •,  0)  can  vanish  identically.  Let 
$i(Mi,  •  ■  •,iii;0,  •  •  • ,  0)  be  developed  into  a  series  of  homogeneous 
polynomials  of  ascending  degrees,  and  let  the  polynomial  of 
lowest  degree, — the  characteristic  polynomial,  as  Mac^NIillan  calls 
it,— be  denoted  by  ip'^'^'^iui,  •  •  •,  m),  its  degree  being  A\.  Mac- 
Millan  considers  the  case  that  the  resultant  R  of  the  characteristic 
polynomials  does  not  vanish.! 

Bliss§  has  also  given  a  treatment  of  this  case  and  has  obtained 
the  result  that,  when  i?  4=  0,  the  number  .Y  has  the  value: 

X  =  n  h. 

1=1 

Another  special  case  of  the  main  theorem,  has  been  investigated 
by  Clements.  II  Let  the  Jacobian  J  vanish  in  the  point  (m;  x) 
=  (0;  0),  and  furthermore  let  Ji  =  J, 

Ji  =  ^7 X  =  0,  (w;  X)  =  (0;  0); 

d{ui,  U2,  ■  •  ■,  ui) 


*  M6canique  celeste,  vol.  1,  p.  72. 

t  Math.  Ann.,  72  (1912),  p.  157. 

X  Cf.  Bliss'.s  critique  of  Poincare's  theorem  and  the  results  obtained  by 
MacMillan,  Transactions  Amer.  Math.  Soc,  13  (1912),  p.  135. 

§L.c. 

II  Bulletin  Amer.  Math.  Soc.  (2),  18  (1912),  p.  451;  Transactions  Amer.  Math. 
Soc,  14  (1913),  p.  325. 


FUNCTIONS   OF   SEVERAL   COMPLEX  VAMABLES.  197 

Jk-i  =     -.. -^  =  0,        (w;  x)  =  (0;  0); 

a(Wl,  ll2,    •  •  ',Ui) 

Then  the  hypotheses  of  the  above  theorem  are  fulfilled,  and 
N=  L 

§  8.    The  Inverse  of  an  Analytic  Transformation 
Let 


(1) 


/T„   =  fn(:Ui,   •  •  •,Un), 

where  fk{ui,  ••■,  Un),  k  =  1,  •••,  n,  is  analytic  in  the  point 
(m)  =  (0)  and  vanishes  there.  If  the  Jacobian  J  of  the  fs  does 
not  vanish,  it  is  well  known  that  the  equations  can  be  solved 
uniquely  for  the  u's  in  terms  of  the  x's,  the  resulting  functions 
being  analytic  at  the  point  (x)  =  (0). 

To  pass  to  the  other  extreme,  if  the  Jacobian  vanishes  iden- 
tically, there  is  a  relation  between  the  /'s.     More  precisely,  let 

T:         1 2ik\  <  h,  k  ^  I,  •  •  •,  n, 

be  an  arbitrary  neighborhood  of  the  point  (u)  =  (0) .  Then  there 
is  a  point  (a)  of  this  neighborhood  and  a  function  F(xi,  •  •  ■ ,  .t„) 
which  is  analytic  in  {x)  at  the  point  {x)  =  (b),  bk  =  fk{cii,  •  •  • ,  fln), 
k  =  1,  •  •  • ,  n,  and  which  has  the  following  property: 

(2)  F{fu  ••.,/„)  ^0, 

where  (u)  is  any  point  of  the  neighborhood  of  (a). 

Thus  the  n  functions /a;(wi,  •  •  •,  Un)  are  connected  by  an  iden- 
tical relation*  (2), 

The  intermediate  case,  that  J  vanishes  at  the  point  (u)  =  (0), 
but  does  not  vanish  identically,  has  been  an  object  of  investiga- 
tion in  recent  years. 

*  Peano-Genocchi,  Calcolo  differenziale  e  integrale,  p.  162.  Bliss,  Prince- 
ton Colloquium,  p.  67,  where  it  is  shown  that,  when  ?i  =  2,  the  point  (a) 
may  be  taken  at  (0). 


19S  THE    MADISOX    COLLOQUIUM. 

First,  let  us  observe,  a  general  solution  of  the  problem  is  given 
by  the  theorem  of  §  7  for  all  transformations  (1)  which  are  such 
that  the  point  (ti)  =  (0)  is  the  only  point  in  this  neighborhood 
which  corresponds  to  (x)  =  (0). 

For  the  case  n  =  2  Clements  discussed  completely  the  above 
transformation  under  this  last-named  hypothesis.  ^loreover, 
his  theorem  cited  in  §  7,  and  Bliss's  results  apply  to  certain  classes 
of  transformations  of  the  kind  under  consideration.  Urner*  and 
Dederickf  have  also  studied  the  problem  from  a  different  point  of 
view, — ^that  of  the  effect  of  the  transformation  on  certain  curves 
which  abut  on  a  point  where  the  Jacobian  vanishes.  Dedericki 
introduced  the  determinant  J2  (§  7)  in  the  case  /  =  2,  and  Urner 
extended  the  definition  to  the  higher  J's. 


*  Transactions  Armr.  Math.  Soc,  13  (1912),  p.  232. 
t  Ibid.,  14  (1913),  p.  143. 
t  Harvard  Thesis,  1909. 


LECTURE   V 

THE  PRIME  FUXCTIOX  ON  AX  ALGEBRAIC  COXFIGURATIOX 

§  1.    The  Algebrmc  Functions  of  Deficiency  1  and  the 
Doubly  Periodic  Functions.     Generalizations 

1.  The  Riemann's  Surface  as  Fundamental  Domain.  The 
algebraic  functions  of  deficiency  unity  and  their  integrals  are 
closely  allied  to  the  doubly  periodic  functions  and  their  related 
functions,  the  theta  and  the  sigma  functions.  It  is  one  of  the 
leading  ideas  which  Riemann  introduced  into  the  theory  and  which 
has  been  further  developed  by  Klein  and  his  school  that  these 
two  classes  of  functions,  from  a  higher  point  of  view,  may  all  be 
considered  as  functions  on  one  and  the  same  foundation  (Sub- 
strat),  the  Riemann's  surface,  idealized  as  a  fundamental  domain. 
Thus  the  ?2-leaved  surface  of  deficiency  1  (or,  more  particu- 
larly, the  two-leaved  surface  with  four  branch  points)  and  the 
parallelogram  of  periods  are,  when  regarded  as  fundamental 
domains  on  which  functions  with  familiar  properties  may  be 
defined,  equivalent. 

The  Theta  Function.  The  single  function  in  terms  of  which 
the  group  of  functions  considered  in  this  theory  can  be  expressed 
is,  when  w^e  make  the  parallelogram  of  periods  and  its  congruent 
repetitions  the  domain  of  the  independent  variable,  the  theta  or 
the  sigma  function.  The  characteristic  properties  of  this  func- 
tion are: 

(a)  that  it  is  single-valued  and  analytic  within  and  on  the 
boundary  of  the  parallelogram; 

(6)  that  its  values  in  corresponding  points  of  the  boundary 
are  related  to  each  other  in  a  simple  manner,  namely, 


a{n  -\-  (jo)  =  —  e  ^     '^cr{u), 

g{u  +  w  )  =  —  c   ^     -  ^(r{u) ; 
199 


200  THE   MADISON   COLLOQUIUM. 

(c)  that  it  has  a  single  root  of  the  first  order  in  the  parallelo- 
gram. 

These  properties  can  be  followed  with  ease  when  the  function 
is  transplanted  to  the  two-leaved  (or  w-leaved)  algebraic  surface 
F,  spread  out  over  the  z-plane.  If  a  system  of  cuts  is  made  in 
F  so  that  a  simply  connected  surface  F'  with  a  boundary  is 
generated,  a  branch  of  the  theta  function  will  be  single-valued 
in  F'.  This  branch  will  be  analytic  in  all  the  ordinary  points 
of  the  surface  and  continuous  in  the  branch-points,  and  its 
values  on  opposite  sides  of  a  cut  will  differ  from  each  other  by 
a  factor  always  finite  and  dift'erent  from  zero.  The  point  oo  plays 
no  exceptional  role.  Finally,  the  branch  in  question  will  have 
a  single  zero  of  the  first  order  in  F'. 

If  the  function  is  considered  on  F,  it  will  be  infinitely  multiple- 
valued.  But  in  the  neighborhood  of  any  point  its  values  can 
be  grouped  to  branches  each  single-valued  there,  analytic  in  the 
ordinary  points,  and  continuous  in  a  branch  point. 

The  Independent  Variable.  I  spoke  above  of  the  single  function 
in  terms  of  which  the  group  of  functions  considered  in  this  theory 
can  be  expressed.  But  a  function  implies  an  independent,  as 
well  as  a  dependent,  variable,  and  the  theta  function  in  the  ordi- 
nary, restricted,  sense  is  simple  because  of  a  felicitous  choice  of 
the  independent  variable.  If  we  follow  this  variable  over  the 
fundamental  domain  taken  now  as  the  w-leaved  surface  F',  we 
find  in  it  a  function  on  this  domain, 

(a)  which  is  everywhere  analytic  in  the  ordinary  points  and 
continuous  in  the  branch-points  and  at  infinity; 

(6)  which  takes  on  boundary  values  differing  by  an  additive 
constant  across  a  cut;  and 

(c)  which  maps  the  neighborhood  of  any  point, — even  though 
this  be  a  branch  point, — on  a  smooth  single-leaved  patch  in  the 
other  plane.  —  The  function  happens  to  be  in  this  case  the  every- 
where finite  integral  of  the  algebraic  configuration. 

Generalizations  for  p>  I.  On  an  n-leaved  algebraic  surface 
of  deficiency  p  >  I  the  algebraic  functions  and  their  integrals 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES.  201 

present  themselves  without  difficulty.  But  they  do  not  yield  a 
transition  to  a  new  fundamental  domain  on  which  a  function 
wdth  the  essential  properties  of  the  theta  function  is  readily 
defined.  Riemann  constructed  functions  in  a  measure  akin  to 
the  elliptic  thetas  by  means  of  the  theta  functions  of  p  arguments. 
But,  aside  from  the  fact  that  his  functions  have  in  general  p 
roots, — never  a  single  simple  root, — they  may  in  particular 
vanish  identically. 

Weierstrass,  on  the  other  hand,  introduced  functions  which 
are  single-valued  and  in  general  analytic  on  F,  but  which  have  a 
finite  number  of  essential  singularities. 

A  way  out  was  found  by  Klein  in  the  use  of  homogeneous 
variables,  already  employed  by  Aronhold  and  Clebsch  in  the 
study  of  transcendental  functions.*  Klein  perceived  still  greater 
possibilities  in  these  ideas  and  carried  through  the  definition  of  a 
function  which,  considered  on  the  allied  manifold  in  the  space  of 
the  homogeneous  variables, — aUied,  I  mean,  to  the  Riemann's 
surface,  F, — is  a  generalization  of  the  elliptic  theta  function, 
namely,  his  yrime  function'^  12(.i-i,  .r2;  yi,  1/2).  The  latter  is  a 
function,  not  of  two  variables  x,  y  corresponding  to  two  points 
of  the  given  algebraic  configuration,  one  of  which,  y,  may  be 
thought  of  as  a  parameter,  x  being  the  variable;  but  of  four 
independent  variables  xi,  X2,  yi,  y^-  It  is  homogeneous  in  Xi,  x-2, 
and  also  in  2/1,  ?/2. 

In  Klein's  investigations  there  are  two  very  distinct  things 
which  he  desires  to  accomplish.  He  wishes,  it  is  true,  to  find  a 
generalization  of  the  elliptic  theta  function.  But  he  also  wishes 
to  obtain  a  function  which  will  formally  be  invariant  of  certain 
hnear  transformations, — often  the  collineations  of  the  space  in 
which  the  basal  algebraic  configuration  (Grundkurve)  is  inter- 
preted. 

To  accomplish  the  latter  end,  the  value  of  the  homogeneous 

*  This  method  was  expounded  systematically  in  Clebsch  and  Gordon's 
Abelsche  Funktionen,  of  the  year  1866. 

tGottingen  lectures  on  the  AbeUan  functions,  S.-S.,  1888,  to  S.-S.,  1889; 
Math.  Ann.,  36  (1889),  p.  1. 


202  THE   MADISON   COLLOQUIUM. 

variables  he  employs  is  unquestioned.  But  in  so  proceeding, 
the  former  object  is  made  secondary, — at  least,  the  homogeneous 
variables  must  be  accepted  from  the  outset,  and  he  does  not 
obtain  in  the  end  a  function  of  a  single  variable  and  a  single 
parameter,  like  the  elliptic  theta  transplanted  to  the  surface  F. 

I  propose  to  give  here  a  direct  solution  of  the  former  problem. 

What  has  all  this  to  do  with  functions  of  several  complex 
variables?  Just  this,  that  the  methods  of  that  theory  yield 
proofs  where  proofs,  in  the  theory  as  developed  by  Klein,  are 
lacking. 

One  word  as  to  the  importance  of  this  mode  of  treatment. 
The  algebraic  functions  and  their  integrals  occupy  a  central 
position  in  analysis  through  their  relation  to  the  geometry  of 
algebraic  curves  and  surfaces,  the  theory  of  linear  differential 
equations  of  the  second  order  with  algebraic  coefficients,  and  the 
automorphic  functions  of  one  and  of  several  variables,  including 
the  periodic  functions  of  several  variables.  The  progress  of 
mathematics  in  the  future,  even  more  than  in  the  past,  will  de- 
pend on  the  rapidity  with  which  the  recruits  can  be  despatched 
to  the  frontier.  As  a  result  of  the  theorems  of  uniformization 
now  rigorously  established  an  improved  treatment  of  the  alge- 
braic functions  and  their  integrals  has  become  possible  and,  by 
reason  of  its  simplicity  and  generality,  appears  suited  to  super- 
cede the  methods  hitherto  used.  In  this  treatment,  the  prime 
function  as  developed  in  the  following  paragraphs  is  the  domi- 
nating factor  and  may  be  made  the  basis  for  the  whole  theory. 

§  2.    The  Prime  Function  Defined  as  a  Limit 

Generalizing  from  the  elliptic  case  considered  by  Aronhold 
and  the  hyperelliptic  case,  which  he  himself  had  treated  at 
length,  Klein  introduced,  for  an  arbitrary  algebraic  configura- 
tion, reduced  by  birational  transformation  to  a  normal  form, 
an  expression  which  he  called  an  "  everywhere  finite  dift'erential," 
and  which  he  writes  as  dux-  It  is  sufficient  for  our  present 
purposes  to  know  that  this  expression  is  analogous,  for  the  neigh- 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARL\BLES.  203 

borhood  of  a  given  point  of  the  configuration,  to  what  would 
appear  in  Weierstrass's  theory  as  x(i)dt,  where  t  denotes  the 
parameter  by  means  of  which  the  neighborhood  in  question  is 
uniformized,  and  x(0  is  analytic  and  does  not  vanish  there. 

Let  P^r,{^)  be  an  Abelian  integral  of  the  third  kind  with  its 
logarithmic  discontinuities  in  the  points  x  =  ^,  x  =  rj,  and  let 

INIoreover,  P^y,ix)  shall  be  so  chosen  that 

-^   fT,  -^  xy' 

Klein  defines  his  prime  function  fi(.ri,  X2;  yi,  yi)  as  the  follow- 
ins:  limit: 


do^xdooye     ''•^ 

dx=0,  dy=Q 

We  can  now  state  the  definition  of  the  prime  function  which 
we  propose  to  develop  in  detail  in  the  following  paragraphs.  Let 
the  algebraic  configuration  be  an  arbitrary  one  of  deficiency 
p  >  1,  and  let  it  be  uniformized  by  automorphic  functions  with 
limiting  circle  in  the  /-plane.*  Let  the  integral  P,  transferred 
to  the  /-plane,  be  wTitten 

Then 

(1)  fi(/,  r)  =     Km     yJAtAre     ''^ 

At=0,  At=0 

In  form,  then,  the  definition  is  identical  with  Klein's.!  But 
whereas  Klein's  dcox  is  single-valued  on  a  homogeneous  configura- 
tion corresponding  to  the  given  algebraic  configuration,  our  dt 
is  not  invariant  of  the  transformations  of  the  automorphic  group. 
Transferred  to  the  surface  F  it  is  infinitely  multiple-valued. 

On  the  other  hand,  Q(t,  r)  is  a  function  of  the  two  independent 
variables  t,  r,  each  being  chosen  arbitrarily  in  the  fundamental 

*  The  details  of  this  uniformization  are  set  forth  in  the  second  edition  of  the 
author's  Funktionentheorie,  vol.  1,  1912,  ch.  14. 

t  Math.  Ann.,  36  (1889),  p.  12.  Cf .  also  Klein's  account  of  the  relation  of  his 
prime  function  to  Weierstrass's  E{x,  y)  and  Schottky's  E{^,  rj);  ibid.,  p.  13. 
15 


204  THE   MADISON   COLLOQUIUM. 

domain  corresponding  to  the  given  algebraic  configuration,  and 
not  on  an  allied  configuration  in  the  space  of  the  homogeneous 
variables.  Here,  r  plays  the  role  of  a  parameter,  t  being  the 
(single)  independent  variable. 

It  is,  however,  important  to  know  the  nature  of  the  depen- 
dence of  12  on  both  arguments,  regarded  as  independent  variables. 
So  far  as  the  analytic  character  of  the  dependence  is  concerned, 
theorems  in  the  newer  theory  of  functions  of  several  complex 
variables  afford  precisely  the  tools  that  are  needed. 

§  3.    The  Existence  Theoreais 

It  is  evident  that,  if  we  are  to  deal  with  such  a  limit  as  the  one 
here  considered  and  infer  the  analytic  properties  of  the  limiting 
function,  it  will  not  suffice  to  study  the  function  P|^  or  Pf/ 
merely  in  its  dependence  on  one  variable  at  a  time.  The  theorem 
of  II,  §  5,  combined  with  the  existence  theorems  as  developed  by 
Neumann,  enables  us  to  obtain  with  ease  the  foundation  needed. 

Let  us  consider  first,  as  lying  nearest  to  the  theory  of  Neumann,* 
an  arbitrary  algebraic  Riemann's  surface  F  and  two  points  ^,  -q 
of  the  same.  Moreover,  ^,  -q  shall  be  ordinary  points,  and  it 
shall  be  possible  to  enclose  them  within  a  circle  K  not  including 
any  branch-point  in  its  interior  or  on  its  boundary. 

Let  ^,  7}  be  joined  by  a  right  line  L,  and  let  F  be  cut  along  L. 
Then  there  exists  a  logarithmic  potential  function  v,  single- valued 
and  finite  in  the  severed  surface,  harmonic  at  all  ordinary  points 
and  continuous  in  the  branch-points  and  at  infinity,  admitting 
harmonic  continuation  across  L  from  either  side,  and  such  that 

Let  d  be  defined  in  the  region  consisting  of  K  cut  open  along 

L  as  follows: 

d  —  arc  (s  —  ^)  —  arc  (s  —  -q) 

=  tan  ^ rr  —  tan  ^ r  , 

X  —  t;  X  —  rj 

-   T  <    d  <   IT, 


*  Abelsche  Integrale,  2d  ed.,  1884,  chs.  16-18. 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES. 


205 


where  z  =  .t  +  yi,  ^'=  ^'  +  ^"i,  v  ^  v'  =  v'  +  v"i-     Then 
(2)  t^=  d+f(x,2j), 

where /(a-,  y)  is  harmonic  throughout  K. 
By  means  of  this  function: 

v  =  v{x,y;^\^";v',v"), 

2p  everywhere  finite  logarithmic  potentials  can  be  constructed^ 
each  admitting  the  modulus  of  periodicity  1  across  one  of  the  2p 
cross  cuts,  but  being  single-valued  across  each  of  the  others. 
In  fact,  let  C  be  a  loop  cut  not  passing  through  a  branch  point. 
Mark  on  C  n  points  ^k,  Jc  =  1,  •  •  •,  n,  so  chosen  that  about  two 
consecutive  points  a  circle  K  can  be  drawn.  The  function  v 
being  formed  for  each  pair  of  consecutive  points,  the  sum  of 
these  n  functions  will  be  an  everywhere  finite  logarithmic  poten- 
tial with  modulus  of  periodicity  2t  across  C. 

These  22?  functions  are  seen  to  be  linearly  independent.  Out 
of  them  a  normal  system  of  p  everywhere  finite  integrals  can 
now  be  constructed: 


(3) 


Ax, 

A2       • 

••     Ap 

B,     •• 

•     Bp 

Wi 

Tri 

0      • 

••      0 

an     •• 

'     aip 

U'2 

0 

iri 

••      0 

^21       • 

•        «2p 

Wp 

0 

0       • 

iri 

flpi     • 

•  •     app 

where  au  =  aa. 

Furthermore,  if  we  denote  the  conjugate  of  the  above  function 
v  by  —  u,  then 

u-]r  vi 

is  an  integral  of  the  third  kind.  Such  an  integral  can  be  obtained 
for  an  arbitrary  pair  of  ordinary  points  ^,  77  of  the  surface  by 
joining  these  points  by  a  curve  L,  interpolating  on  L  n  —  1 
points  so  that  two  consecutive  points  lie  in  a  circle  K,  forming 
the  foregoing  function  for  each  pair  of  consecutive  points,  and 
adding  these  n  functions  together. 


206 


THE   MADISON   COLLOQUIUM. 


Finally,  this  integral  can  be  reduced   to  a  normal  integral 
n|,(z)  with  vanishing  moduli  of  periodicity  across  the  A  cuts: 


(4) 


n.Jz) 


Ai 


0 


Bi 


B. 


2ivt>  ' 


.••       0     2^!"     •• 

where  iOk(z)  denotes  a  branch  of  the  function  taken  in  the  simply 
connected  surface  F',  and 

(5)  wt  =  iVkiO  —  Wk(v)- 

The  integral  U^ni^)  is  completely  determined  save  as  to  an 
additive  constant,  which  is  any  function  of  |,  rj. 
If  we  set 

(6)  ni;"  =  n^,(2)  -  u^,(io), 

then* 

(7)  n|;  =  nf:^. 

The  scheme  of  the  moduli  of  periodicity  of  the  function  11^^,  when 
regarded  as  a  function  of  one  variable  at  a  time,  is  as  follows. f 


A,     . 

•     Ap 

Bi 

•        Bp 

X 

0     . 

•      0 

2ui^     •  • 

2wp 

y 

0      • 

.      0 

-2tvp     '  • 

•     -  2wp 

^ 

0      • 

.      0 

2w7     •  • 

2tv;' 

V 

0      • 

•      0 

-22V?       '• 

■     -  2wl' 

(8) 


In  a  similar  manner,  the  normal  integral  of  the  second  kind 
is  obtained  :| 

(9)  Z^iz)  =  ^-^  +  mz), 

where  ^  is  an  ordinary  point  and  21(2)  is  anal}i;ic  at  s  =  ^. 


(10) 


ZM 


Ai 


0 


0 


Br 


Br 


-  2<pm 


-  2<pp{0 


*  Cf.  Appell  et  Goursat,  Fonctions  algdbriques,  p.  327. 
t  In  these  formulas,  x,  y  denote  complex  variables. 
X  Neumann,  Abelsche  Integrate,  2d  ed.,  1884,  p.  206. 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARL\BLES.  207 

where  (pk{'z)  denotes  the  integrand  of  the  normal  integral  Wk{z): 

'>^k{z)  =   I    <pk{z)  dz  +  const. 

§  4.  Dependence  on  the  Parameter 

The  function  ^  of  §  3,  (2)  has  hitherto  been  considered  solely 
in  its  dependence  on  x  and  y.  It  has  the  following  further 
property,  as  is  seen  directly  from  the  existence  proof.  Let  K' 
be  a  circle  concentric  with  K  and  of  smaller  radius,  and  let  the 
points  ^,  77  be  restricted  to  the  interior  of  K'.  Then  /(.r,  y), 
which  now  becomes  a  function  of  i',  ^",  -q',  7]"  as  well  as  of  x,  y, 
remains  finite  when  {x,  y)  ranges  over  K  and  (^',  ^"),  (r)',  -q") 
range  independently  over  K',  provided  that  we  complete  its 
definition  by  demanding,  for  example,  that  it  vanish  in  a  fixed 
point  of  K  exterior  to  K'.  The  function  is  defined  only  when 
the  points  (^',  ^"),  (77',  t)")  are  distinct. 

It  is  now  readily  inferred  from  the  well-known  properties  of 
the  logarithmic  potential  that  the  function  conjugate  to  f{x,  y) 
also  remains  finite  when  {x,  y),  {^',  ^"),  (77',  -q")  vary  as  above, 
the  definition  of  this  function  being  completed,  for  example,  by 
demanding  that  it  vanish  in  the  same  fixed  point  of  K. 

Finally,  it  is  seen  that  v  remains  finite  in  the  part  of  F  exterior 
to  K  when  ^,  -q  range  over  K'.  Hence  a  branch  oi  u  -\-  vi,  con- 
sidered in  a  simply  connected  region  S  of  F,  remains  finite  there, 
provided  ^,  -q  are  exterior  to  S  and,  moreover,  that  their  minimum 
distance  from  the  boundarj^  of  S  does  not  fall  below  a  certain 
positive  constant.  Similarly,  the  moduli  of  periodicity  of 
w  +  ^  i  across  the  A  and  the  B  cuts  remain  finite  when  ^,  77  range 
over  K'. 

Let  ^oj  ^0  be  two  distinct  points  of  K' ,  and  let  zq,  ioq  be  two 
ordinary  points  of  F  distinct  from  ^0,  ^o-  Consider  the  cylindrical 
region  T  =  {T^,  Tr,,  T^,  Ty)  corresponding  to  small  circles  about 
each  of  the  points  ^0,  ^0,  20,  wq.  If  three  of  the  four  variables 
are  assigned  arbitrary  values  in  their  circles  and  then  held  fast, 


208  THE   MADISON   COLLOQUIUM. 

while  the  fourth  is  allowed  to  range  over  its  circle,  it  appears 
from  (6)  and  (7)  that  Ul'^  is  analytic  in  this  variable  alone. 
Furthermore,  from  the  considerations  which  have  just  preceded, 
it  is  seen  that  when  all  four  variables  range  over  their  circles, 
n|^  remains  finite  in  T.  We  infer,  then,  from  the  theorem  of 
II,  §  5,  in  its  restricted  form  that  11  is  analytic  in  all  four  variables 
regarded  as  simultaneous. 

Next,  let  us  consider  the  function  11  when  Zo  =  ^o,  the  point  z 
lying  in  the  circle  about  ^o-  The  points  ?o,  Vo,  wo  are  distinct 
ordinary  points.  But  it  is  necessary  now  to  demand  that  s  shall 
not  coincide  with  ^.     If  w^e  write 

(11)  ni;"  =  log  (z  -  ^)  +  51(2,  w,  ^,  v), 

then  51  is  defined  at  all  points  of  T  except  those  of  the  locus 
2  =  ^,  and  51  is  finite.  It  follows  here,  as  in  the  earlier  case,  that 
51  is  analytic  in  those  points  of  T  in  which  it  is  defined.  And 
now  comes  a  typical  application  of  the  theorem  of  III,  §  4, 
relating  to  removable  singularities.  From  it  we  infer  that  51 
approaches  a  limit  in  each  of  the  excepted  points,  and  that,  if 
51  is  defined  there  as  equal  to  its  limit,  then  51  will  be  analytic 
there. 

Similar  formulas  hold  for  other  coincidences  of  the  points 
Zo,  w'o,  ^0,  VO'     Thus,  when  all  four  points  coincide, 

(12)  Ui;  =  log  [^  I  ^^ll'^  I  ^^)  +  A(z,  w,  ,^  7?), 

where  A  is  analytic  in  all  four  arguments,  regarded  as  simul- 
taneous, in  the  point  in  question. 

§  5.    The  Functions  in  the  Automorphic  Fundamental 

Domain 
We  proceed  now  to  transfer  all  the  functions  from  the  ?i-leaved 
Riemann's  surface  of  the  z-plane  to  a  fundamental  domain  % 
in  the  unit  circle  of  the  ^plane.     The  relation  between  z  and  t 
shall  be  expressed  by  the  equation 

(13)  z  =  (pit),     or     /  =  a;(~). 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES. 


209 


The  fundamental  domain  ^  for  the  automorphic  group  G  in 
question  can  be  chosen  as  a  circular  polygon  of  42?  sides  joined 
in  pairs.  In  the  canonical  system  of  cuts  in  the  7i-leaved  surface 
the  positive  direction  of  a  B-cut  shall  be  oriented  to  the  positive 
direction  of  an  ^-cut  as  the  positive  axis  of  ordinates  to  the 
positive  axis  of  abscissas;  and  the  left  bank  of  each  cut,  when 
the  latter  is  described  in  the  positive  sense,  shall  be  taken  as  the 
positive  bank.  The  four  banks  appear,  then,  in  the  ^figure  as 
indicated.  We  shall  regard  the  points  of  the  curves  A~,  B~ 
as  pertaining  to  jv;  those  of  A^,  B^  as  not  pertaining  to  '^. 


Fig.  4. 


Fig.  5. 


The  Normal  Integral  of  the  Third  Kind.  In  building  up  the 
integral  of  the  third  kind  we  have  excluded  the  case  that  f,  rj 
lie  at  a  branch  point  or  at  infinity.  Such  points  present  no 
peculiarity  in  the  ^-figure.  We  can  remove  this  exception  in 
either  one  of  two  ways :  (a)  we  can  go  back  and  extend  the  earlier 
considerations  to  the  cases  excepted,  or  (6)  we  can  establish  the 
existence  theorems  directly  for  the  case  of  the  fundamental 
domain  i^. 

There  is  no  difficulty  in  carrying  through  the  first  method. 


210  THE   MADISON   COLLOQUIUM. 

The  second  method,*  however,  yields  the  desired  results  with 
essentially  the  same  machinery  as  that  used  for  the  n-leaved 
surface,  but  with  no  excepted  cases  whatever. 

The  final  result  in  either  case  is  that  the  formulas  for  which 
(11)  and  (12)  are  representative  hold  unchanged  for  the  ^-plane. 

Thus,  for  example,  if  t,  r,  f,  t'  be  any  four  distinct  points  of 
%,  we  shall  havef 

(14)         -n^r,,  =  log  ^l^Z^')^lrZJ't]  +  ^(^^  ^;  ^'^  ^'), 

where  %{t,  r;  t',  t')  is  analytic  in  its  four  argument3  throughout 
the  cylindrical  region  {%,  %,  ^,  %)  inclusive  of  all  boundary 
points. 

The  Transformations  of  the  Functions.     Let 

(15)  t,=  La.{t),  a  =  I,  ■■','p, 

be  the  linear  transformation  of  G  which  carries  A~  into  A^', 
and  let 

(16)  ip+a  =  Lp+^{t),  a  =  1,  •••,Vy 

carry  Br  into  B^.     The  inverse, 

t'  =  L-\t), 

will  carry  Aa!^  into  A~;  and  similarly  for  (16). 

The  transformations  (15),  (16)  together  with  their  inverses 
constitute  a  system  of  generators  for  the  automorphic  group  G. 

Any  single-valued  function  on  F  which  has  no  other  singularities 
than  poles  goes  over  into  a  single-valued  function  of  t  having  no 
other  singularities  than  poles  within  the  unit  circle  |  <  |  <  1  and 
(with  the  single  exception  of  a  constant)  having  the  circumference 
of  the  circle  as  a  natural  boundary. 

The  Abelian  integrals  of  the  first  and  second  kind  also  go  over 
into  single-valued  functions  of  t  having  no   other  singularities 

*  Cf.  Fricke-Klein,  Automorphe  Funktionen,  vol.  2,  chap.  1. 

t  The  notation  for  the  dependent  variable  here  and  in  the  following  is  the 
same  whether  the  arguments  are  taken  in  the  i-circle  or  on  the  a-surface.  It 
will  be  clear  from  the  context  each  time  which  is  meant. 


FUNCTIONS   OF    SEVERAL   COMPLEX   VARIABLES.  211 

than  poles  within  the  circle.     Those  of  the  third  kind  have 
logarithmic  singularities. 

Consider,  in  particular,  a  normal  integral  of  the  first  kind, 
w'jj,  where  t,  r  correspond  respectively  to  z,  w.  Then,  if  the 
transformation  (15)  be  performed  on  t, 

[  wi-'  =  w'J,  k  ^  a\ 

(17) 

I  w*y  =  wl''  +  iri. 

Corresponding  to  a  transformation  (16)  we  have 

(18)  tti"-"'' '  =  w'J  +  o,„. 

If,  on  the  other  hand,  r  is  transformed  and  t  held  fast,  the 
sign  of  the  additive  term  is  reversed. 

It  appears,  then,  that  the  scheme  (3)  applies  to  the  integrals 
w^^,  k  =  1,  ■  •  -,  p,  when  regarded  as  functions  of  t  alone,  the 
transformations  being  those  of  (15),  (16).  When  these  integrals 
are  regarded  as  functions  of  r  alone,  each  term  in  the  scheme 
(3)  is  replaced  by  its  negative. 

The  scheme  (8)  applies,  with  the  requisite  changes  in  the 
letters,  to  the  transformed  integral  H'J^,. 

Homomorphic  Functions.  Any  Abelian  integral  u,  when  sub- 
jected to  a  transformation  of  the  group  G,  experiences  a  trans- 
formation of  the  type 

(19)  u'  =  u  +  A, 

where  ^  is  a  constant.     These  functions  come  under  the  general 
class  of  functions  which  undergo  a  transformation  of  the  form 

(20)  (H'SI)'  J^«-^^  +  0, 

where  M,  •  •  • ,  Q  are  constants.     The  general  class  of  functions 
which  have  this  property  are  called  homomorphic  functions* 

To  the  algebraic  functions  on  F  correspond  functions  u  of  t, 
for  which   (20)  reduces  to  the  identity.     These  functions  are 

*  EHein,  Ueber  lineare  Different ialgleichungen  zweiter  Ordnung,  Gottingen, 
S.-S.,  1894  (lithographed),  p.  492. 


212  THE   MADISON   COLLOQUIUM. 

absolute  invariants  of  the  group  G,  and  are  automorphic  func- 
tions of  t. 

§  6.    An  Auxiliary  Function 

Let  t,  T  be  any  two  distinct  points  of  the  fundamental  domain 
5.  We  can,  without  loss  of  generality,  think  of  them  as  interior 
points,  since  the  precise  boundary  of  5  can  at  any  point  P  be 
modified  sHghtly  by  what  is  known  as  an  "  erlaubte  Abande- 
rung,"  the  choice  of  the  boundary  as  circular  arcs  being  made 
merely  for  simphcity  and  definiteness. 

Let  t'  =  t  -\-  At  and  r'  =  r  +  Ar  be  two  variable  points  of 
the  neighborhoods  respectively  of  t,  r.     Form  the  function 

X'(^  T,  A^,  Ar)  =  A/ Are     '''^''  '^^\ 
Then,  by  (14): 
X'(<,  r,  Af,  Ar)  =  (^  -  r  -  Ar)(^  -  r  +  AOe''~"^''"=  '+^'' ^+^^\ 

But  the  function  5((^,  r,  t' ,  r')  is  analytic  in  the  point  {t,  r,  t,  r), 
and  hence  X'(i,  r,  M,  Ar)  approaches  a  hmit  when  A^,  Ar  inde- 
pendently approach  0  as  their  limit.     We  have,  then, 

lim     X'(^,  r,  M,  Ar)  =  (^  -  r)'e'^'--'^'' '' *'  '^  =  X(^  r). 

The  excepted  points,  t  =  t,  are  seen  to  be  but  removable 
singularities  for  the  function  X,  and  thus  X{t,  r)  is  defined  and 
is  analytic  throughout  the  cylindrical  domain  {%,  %).  If  we 
regard  r  as  a  parameter,  X,  considered  as  a  function  of  t,  van- 
ishes twice  in  the  point  t  =  r  and  nowhere  else  in  %. 

To  obtain  the  function  X(^,  r)  throughout  its  entire  domain 
of  definition  we  could  use  the  same  limiting  process  as  above, 
subjecting  t  and  r  independently  to  transformations  of  G.  It 
is  sufficient,  however,  to  know  how  X  behaves  on  the  boundary  of 
1^,  or  {%,  5).  In  order  to  ascertain  these  relations,  we  will 
employ  the  method  which  is  familiar  when  one  is  dealing  with 
fundamental  domains.* 


Osgood,  Funktionentheorie,  vol.  1,  chap.  10,  §  8. 


FUNCTIONS   OF  SEVERAL   COMPLEX   VARL^BLES.  213 

The  Behmior  of  X(t,  r)  on  the  Boundary.  Let  r  be  an  interior 
point  of  g,  and  let  c  be  a  point  of  B~,  c^  its  image  on  B^: 

(21)  c,  =  Lp+a(c). 

Then,  by  (8), 

=  n,.,(^')  -  n.,(T')  +  2uC. 

Let  c'  =  c  +  Ac  be  a  second  point  of  B~  near  c,  c'^=  c^-\-  Ac„ 
its  image  on  B^.  We  have,  then,  by  the  aid  of  the  last  equa- 
tion, and  (18),  (8): 

nx  =  n.,(c:)  -  n..(r')  +  2iv'}^' 

=  WJ,,,  +  2w7  +  2wr^'  +  2fl„„. 
Turning  now  to  the  functions 

X(c„,  T,  Ac„,  At)  =  Ac„ATe~"^£', 

X(c,  r,  Ac,  At)  =  AcAtc"''"''^' 
it  is  seen  that 

X(c„,  T,  Ac„,  At)  =  ^^  c-^'^'-'f ''-'"- X(c,  t,  Ac,  At). 

Let  Ac,  At  approach  0  as  their  limit.     Then 

(22)  X(c.,t)=^c-^^-^-"-X(c,t). 

If  c,  Ca  had  been  chosen  on  A~,  A^,  where  now 

c,  =  X,(c), 

we  should  have  had,  corresponding  to  (22) : 

(23)  X(c.,  T)=-^  Xic  r). 

The  Monogenic  Analytic  Function  X(t,  t)  and  its  Transforma- 
tions under  the  Group  G.  We  are  now  in  a  position  to  show  that 
X(t,  t)  can  be  continued   analytically  throughout   the   domain 


214  THE  MADISON   COLLOQUIUIVI. 

(K)  =  (K,  K),  when  K  refers  to  the  unit  circle.  Since  the  details 
are  precisely  parallel  to  those  in  the  treatment  of  the  (r-function, 
Funktionentheorie,  Chap.  10,  §  8,  they  may  be  left  to  the  reader. 
We  thus  obtain  a  function  analytic  throughout  this  domain  and 
not  admitting  analytic  continuation  beyond  it.  This  function 
has  the  following  propetties. 

(i)  X{t,T)  =  X{r,t); 

(ii)  X(/,  r)  =  (t-  ry-nt,  r), 

where  '^{t,  r)  is  analytic  throughout  {K)  and 

^(r,  r)  +  0; 

(iii)  X(^„,  r)  =  L'^(t)X{t,  t),  a  =  1,  •  •  •,  p; 


(iv)  X(Wa,  r)  =  L'^^M  e-  -"'""Xa  r),    ,,  =  1,  . . .,  p; 

(v)  X{t,T:)  =  K{T)M,r); 

(vi)         Xa  i^„(T))  =  L'^Ur)  e<^-"'-'^X{t,  r). 

If  the  transformation 

t^  =  L^{i),  q;  =  1,  ••  •,  p, 

is  replaced  by  its  inverse, 

t'  =  irHO  =  ^lait), 

formulas  (iii)  and  (v)  become: 

(vii)  x(ZrKO,  r)  =  J/Koxa  r), 

(viii)  X{t,L-\T))  =  M:iT)X(t,T), 

^^"^^^  =  L'ALrHt)]  • 

But  if 

tp+a  =  Lp+Jf),  a  =  1,  •  •  •,  p, 

is  replaced  by  its  inverse, 

t'  =  i;+a(0  =  Mj^M, 
formulas  (iv)  and  (vi)  are  replaced  by  the  following: 


FUNCTIONS  OF  SEVERAL  COMPLEX  VARIABLES.      215 

(ix)  X(L-Ut),  r)  =  M^Ut)  e'\'-'^'^'^X(t,  r), 

(x)  X(^,  L-Ur))  =  Ml,Ur)  e-'-'J-^'^'^'^M,  r) , 

It  will  be  observed  that  Li(t),  L'pj^JJ),  a  =  1,  •••,  p,  are 
analytic  and  different  from  zero  throughout  K. 

§  7.    The  Prime  Function  Q(t,  r) 

The  prime  function  ^(t,  r)  is  now  defined  as  follows.*  The 
two  square  roots  of  the  function  X{t,  r)  can  be  so  grouped  as  to 
yield  two  functions  each  single-valued  and  analytic  throughout 
{K).     Let  one  of  these  functions  be  denoted  by  Vx(^,  r).     Then 

{A)  Q{f,  t)  =  CVXOV), 

where  C  denotes  a  constant  not  zero. 

This  function  has  the  following  characteristic  properties,  which 
are  easily  proven  from  its  definition  and  the  developments  of 
the  last  paragraph. 

(a)  Q.{t,  t)  is  analytic  in  the  cylindrical  region  {K)  =  {K,  K) 
and  has  the  boundary  of  this  region  as  a  natural  boundar3^ 

(6)  n{t,  t)  =  -  12(r,  /). 

(c)  mr)  =  (.t-T)Q(,t,r), 

where  Q{t,  r)  is  analytic  in  (K)  and 

More  generallv,  let 

f  =  L{t) 


*  In  the  definition  of  X(;,  t)  it  is  not  essential  that  the  normal  integral  I1Jt> 
be  used.  Any  other  integral  of  the  third  kind,  PJt',  which  permits  the  inter- 
change of  parameters  and  arguments  would  serve  equally  well.  Thus  a 
prime  function  flp(^  t)  would  result  which  has  the  same  properties  (a),  (6),  (c), 
but  for  which  (di)  is  replaced  by  a  pair  of  similar  equations,  each  right  hand 
side  having  an  exponential  factor  whose  exponent  is  a  linear  function  of 
ti'l''-,  •  •  • ,  Wp*-'. 


216  THE   MADISON   COLLOQUIUM. 

be  any  transformation  of  G.     Then 

W,  r)={t-  r')  Q'{t,  r), 
where  Q,'{t,  r)  is  anal}i;ic  in  {K)  and 

Q'{i,  t)  +  0. 


m.,  r)  =  Vl:  (t)  Q(t,  t)  ; 
(a)  1  , 

Here,  each  square  root  denotes  a  function  of  t  analytic  in  K  and 
different  from  zero  there.  The  vahie  of  these  functions  is  given 
below. 

Furthermore 


id') 


Q(t,T^)    =<ljjT)^{t,T), 


[^{t,  Tp+J  =  Vz;^„(r)  e'K'-'^a.M  it,  t), 


where  the  square  roots  denote  the  same  functions  as  above,  in  (d) . 
These  square  roots  have  the  following  determination.     The 
transformations 

t'  =  LSt),     f  =  L^Ut) 

are  hyperbolic.     Let  them  be  written  in  the  form 
7, y=^-. T'  0  <  ^. 

t    ~  S  1  f  ~~  S  1 

The  fix  points  i'o,  ti  lie  on  the  circumference  of  the  fundamental 
circle.     Then 

*  —  Si 
where  L{t)  is  any  one  of  the  functions  L^{t),  Lp+^{t). 

§  8.    The  Deteemination  of  ^(t,  r)  by  Functional 

Equations 

The  properties  (a),  (6),  (c),  (d)  or  {d')  serve  to  characterize  the 
function  1]  completely.     For,  let  "^{t,  r)    be  a  second  function 


FUNCTIONS   OF   SEVERAL   COMPLEX   VARIABLES.  217 

having  the  same  properties.  Let  r  be  an  arbitrary  point  of  5> 
and  let  it  be  held  fast.     Form  the  function 

^g  t) 
m,  r)  ' 

Then  this  function,  regarded  as  a  function  of  t,  will  be  analytic 
in  K  except  for  removable  singularities  in  the  points  t  =  r  and 
in  the  images  of  this  point  under  the  group  G.  Let  it  be  defined 
in  these  points  as  equal  to  its  limit.  The  new  function  is  analytic 
without  exception  in  K,  and  does  not  vanish  there. 

From  {d)  it  follows  further  that  this  function  is  in\'ariant  of 
the  transformations  of  G.  It  is,  therefore,  a  constant,  as  can  be 
seen  at  once  by  transforming  it  to  the  ?i-leaved  surface  F.     Hence 

-^  =  fir),         nt,  T)  =  fir)  Qit,  r),        fir)  +  0. 

This  last  relation  is  an  identity  in  t,  r,  and  hence  can  equally 
well  be  written  in  the  form 

■^ir,  t)  =  fit)  fi(T,  t). 

Now  apply  the  property  represented  by  (6).     It  follows  that 

-■^it,r)=  -  fit)  Qit,  r). 
Hence 

fit)  =  fir), 

and  this  completes  the  proof. 

From  the  foregoing  result  it  is  seen  that  the  properties  (a), 
•  •  • ,  id)  can  serve  as  the  basis  for  an  independent  definition 
of  the  prime  function  Qit,  r).  Thus  the  function  might  be 
represented  by  an  infinite  product,  as  Weierstrass  defined  his 
elliptic  cr-f unction.  And  just  as  AYeierstrass  made  the  latter 
the  basal  function  for  the  whole  theory  of  the  elliptic  functions, 
so  the  algebraic  functions  of  deficiency  p  >  1,  and  their  integrals, 
can  be  represented  in  terms  of  Qit,  r).  We  proceed  to  give  the 
fundamental  formulas. 


218 


THE   MADISOX   COLLOQUIUM. 


§  9.    The  Abelian  Integrals  in  Terms  of  the  Prime 

Function* 

The  Functions  11^,. (0,  H^V-     From  (c)  and  {d)  of  §  7  it  is  seen 
that  the  formula 

^{t,  a) 


n..(0  =  log 


m,  r) 


gives  a  particular  normal  integral  of  the  third  kind,  the  general 
integral  differing  from  the  above  by  an  additive  term  which  is 
an  arbitrary  function  of  a,  r.  In  the  absence  of  any  reason  to 
the  contrary  we  set  this  term  equal  to  zero,  i.  e.,  we  lay  down 
arbitrarily  the  definition  1.  Thus  this  integral  is  completely 
determined. 


If  we  set 


U  =  L^{t),     tp+a  =  Lp+a{t),        a  =  1, 


P, 


dt       ^«' 


dt 


P+a 


dt 


t 


P+a> 


we  find 


n..^(^)     =  n,,(o  + 1  log  (rl, 

U,^^^,M  =  n,,(/)  +  i  log(r;^„  +  2iC  -  a,.', 


n. .. ,  (0  =  n,,(o  -  i  log  t;+,  -  2u'r  +  a. 


II 


The  function  n^'.  is  now  represented  as  follows: 

Q(s,a)n(t,T) 

n^,  =  log 


0(5,  r)  Q{t,  a) ' 

For  this  function,  regarded  as  a  function  of  s  alone,  the  formulas 
(yli)^hold  without  modification.     When  t  is  the  sole  independent 

*  The  deductions  of  this  paragraph  are  suggested  by  corresponding  formulas 
in  the  case  p  =  1,  the  function  il{t,  t)  corresponding  to  d{t  —  t)  or  ff{t  —  t), 
Cf.  Klein,  Math.  Ann.  36  (1889),  p.  11. 


FUNCTIONS   OF  SEVERAL   COMPLEX   VARIABLES. 


219 


variable,  the  second  term  on  the  right  of  the  second  formula  in  (^i) 
is  reversed  in  sign. 

On  the  other  hand,  the  formulas  {A2)  are  simplified. 


(^2') 


n  '  =  n  ' 


n;;^^, ,  =  n;;  +  2<' 


The  new  formulas  (As)  are  similar,  the  last  term  being  reversed 
in  sign. 

The  Functions  w'J,  $a(0-     From  {A^')  and  II.  we  obtain: 


III. 

or 


,/'  =  1 


^{s,  Cp+„)  12(f,  c) 


w„  =  ^  log  f^-:x7^7:r^: — t  =  2  log  -7^77^    +  const. 


fi(f,  c)fi(a,  Cp+„) 


^{t,  c) 


The  expression  on  the  right-hand  side  is  multiple-valued,  but 
the  different  values  can  be  grouped  so  as  to  yield  single-valued 
functions  each  analytic  in  K.  We  choose  that  one  of  these 
functions  which  vanishes  when  t  =  s,  ot  t  =  a. 

A  second  expression  for  w'J  as  the  integral  of  a  single- valued 
function  is  given  below.  Formula  VII. 

Let  ^Jf)  be  defined  as  follows: 


^a{t)   = 


dt 


Then 
IV. 


$.(0  =  1^^  log 


^{t,  Cp+^) 

n(t,  c)    ' 


(t)  dt. 


The  Functions  Y^it),  Y'l',    Let 

V.     F,(0  =  ^  u^rit)  =  -  |:  log  n{t,  r)  =  ^-^  +  m,  t). 

If  we  differentiate  (^1)  with  respect  to  r,  we  obtain 

I       ^'^^"^  ^  ^'^^^' 
16 


220 


THE   aiADISOX   COLLOQUItTM. 


If,  furthermore,  we  differentiate  (^3)  with  respect  to  r,  we  have 


(^2) 


'p+a  'p-ra.  T, 


■p-ra' 


p-\-a 


where  the  accents  denote  differentiation. 
Let  Ft'  be  defined  as  follows: 

ir  =  Y^(s)  -  Y^it). 

For  this  function,  regarded  as  a  function  of  s,  formulas  (£1) 
hold  unchanged.  TMien  t  is  taken  as  the  sole  variable,  the  last 
term  in  these  formulas  is  reversed  in  sign. 

On  the  other  hand,  (^2)  is  replaced  by 


(B,') 


ytt  __  y»«  yt       __     ^     yst 


From  (Bi)  it  follows  that 
VL  ^a{T)  =  ^{YM-  F,(W«)}. 


VII. 


iv'J=hf{YM-  i;(c^JWr. 


The  Derivatives  of  ^{t,  r).     Let 
"i(^>  r)  = 


%{t,  t)  = 


dQ{t,T) 


dt       ' 
From  V.  it  follows  that 
VIII.  9.,{t,r)  =  -  Qit,r)Y,(t). 

From  (h)  we  have: 

Qi{t,  t)  =  -  Qiir,  t). 


Hence 
IX. 


^lit,  r) 


9it,r)Y,(T). 


FUNCTIONS  OF  SEVERAL  COMPLEX  VARLiBLES.      221 

The  Functions  Y'~^\t).     From  the  relations 
dr  (t-  ^)2-t--iil^.  ^>'> 


d--'Y.(t)       On -1)1 

where  %m-i(t,  t)  is  analytic  within  and  on  the  boundary  of  (^5,  %), 
we  are  led  to  define  Y^:^''\t)  as  follows: 

■'       (m  —  1) !     c5t"*  1 
Here, 

From  (5i)  we  obtain  the  formulas: 

c    yrco  =  YT\t), 


iC) 


YT\tv+.)  =   Y^rXt)  -  (,„  !  i^;e-^Xr). 


10.  The  Integral  of  the  Second  Kind  on  F 


The  function  Y^{t),  when  transferred  to  the  ?i-leaved  surface 
F,  is  an  integral  of  the  second  kind  with  a  simple  pole  and  with 
its  moduli  of  periodicity  corresponding  to  the  yl-cuts  all  zero. 
It  differs,  however,  from  the  integral  there  taken  as  the  normal 
integral,  namely  Z^(z),  as  follows. 

We  denoted  by 

Z  =   (p{t) 

the  function  which  maps  the  n-leaved  bounded  surface  F'  on 
the  fundamental  domain  %.  Let  r  be  a  point  of  %  corresponding 
to  an  ordinarj^  point  ^  of  F'.     Then 

^^(^)  =  ^4-^ + 21(2),     YAt)  =  ^  +  m), 

where  2t(2)   and   5B(^)   are  analytic  respectively  in   the  points 

z  =  ^  and  t  =  T. 


222  THE   M.iDISON   COLLOQIHUM. 

It  follows,  then,  since 


Z-^=  cp'(r)(t  -  r)  +  -^  (t  -  rf  + 


91 


and  <p'{t)  =!=  0,  that 

XL  Zf(z)=^}\(0+/(T). 

if   (T) 

This  last  result  is  fundamental  in  showing  the  unfortunate 
choice  made  in  the  earlier  case  in  defining  the  normal  integral 
of  the  second  kind,  whenever  it  is  a  question  of  the  dependence 
of  this  integral  on  the  parameter.*  As  ^  approaches  a  branch 
point  of  F,  the  integral  Z^{z)  is  seen  to  approach  no  limit,  and 
its  moduli  of  periodicity  across  the  5-cuts, — the  integrands  of 
the  everywhere  finite  integral  on  F, — or  certainly  some  of  them, 
become  infinite.  Nevertheless  Z^{z)  has  a  perfectly  definite 
value  when  ^  lies  in  a  branch  point,  and  the  relation  of  the  integral 
in  this  case  to  Y^it)  is  obtained  from  the  formulas 


c^(«)(r) 


m\ 


{t-  rr+  ■•',  <p^^\r)  +0, 


Xr.  Z^{z)  =  y  ^-(,^  j      YAt)  +  const. 

There  is,  then,  complete  discontinuity  in  the  dependence  of 
Z^{z)  on  ^  at  a  branch  point,  and  this  discontinuity  does  not 
correspond  to  any  important  property  of  the  integral  of  the  second 
kind.  On  the  other  hand,  the  points  r  for  which  (p'{t)  =  0  are 
in  no  wise  exceptional  points  for  the  function  Y^{t). 

It  appears,  then,  that  the  earlier  integral  should  have  been 
normalized  so  as  to  correspond  to  Yj{t).  This  can  be  attained 
without  reference  to  Y^{t)  by  replacing  the  earlier  Z^{z)  by 


*  The  present  modification  is  analogous  to  the  one  introduced  by  Klein 
through  the  use  of  his  "  everywhere  finite  "  differential  dwi. 


FUNCTIONS   OF  SEVERAL   COMPLEX   VARIABLES.  223 

Nor  is  this  the  only  point  at  which  this  factor  enters.  We 
turn  next  to  the  functions  which  correspond  to  the  adjoint  C„_3, 
namely,  the  integrands  of  the  everywhere  finite  integrals,  and 
we  shall  find  the  factor  reappearing  there,  also.  In  §  14  we  shall 
discuss  at  length  the  nature  of  this  factor. 

§  11.    The  Integrands  of  the  Integrals  of  the  First  Kind 
If  we  write  u\(z),  considered  on  the  surface  F,  in  the  form 

and  recall  that,  when  Wa_  is  considered  in  K, 

lol"  =    I  ^M  dt  +  const., 
we  see  that 
XII.      <pM  =  ^)  *a(0,        or        <pM  j(^^  =  $„(0. 


The  function  (Pa,{z)  is  single-valued  on  F,  and  its  only  singu- 
larities are  poles,  which  are  situated  in  the  branch  points  of  F. 
^a(t),  on  the  other  hand,  is  analj^tic  without  exception  in  ^, 
or  in  K.  Here  again,  then,  it  is  the  factor  ll{dtldz)=(p'(t)  that 
removes  the  singularities  from  an  earlier  function. 

The  functions  ^aii)  are  not  invariants  under  the  group  G, 
nor  are  they  even  homomorphic.  They  take  on  a  factor  which 
is  finite  and  different  from  zero.     In  fact, 

(D)        **(^«)  =  z:(o'    ^^^^^-^»)  =  zj^' 

where  k  =  1,  •  •  • ,  p  and  a  —  1,  •  •  • ,  p. 

In  the  treatment  of  Abel's  Theorem  and  the  theorem  of 
Riemann-Roch,  when  the  functions  entering  were  considered  on 
F,  there  were  exceptional  cases  that  required  special  consider- 
ation, due  to  a  function's  becoming  infinite.  When  the  same 
theorems  are  treated  on  the  fundamental  domain  %  and  the 
method  of  contour  integration  is  used,  the  results  are  completely 
general,  no  exceptions  whatever  occurring. 


224  THE   MADISON   COLLOQUIUM. 

Xoether's  normal  curve  C  is  given  by  the  equations : 

pXa  =  ^a{t),  a=  1,   ••■,  V- 

Tlie  hyperelliptic  case  being  excluded,  the  curve  has  no  multiple 
points.  The  treatment  by  means  of  the  functions  $a(0  is 
simple  and  complete. 

By  the  aid  of  these  functions,  too,  a  canonical  Riemann's 
surface  which  Klein*  has  introduced  can  be  treated  satisfactorily, 
analytic  proofs  replacing  the  customary'  algebraic  assumptions. 
Klein  projects  the  curve  C  on  a  pencil  of  planes,  i.  e.,  he  sets 

z  =  — , 
where 

the  UkS  and  Vk^  being  non-specialized  constants.  As  the  other 
variable,  s,  he  takes  the  folloT\dng: 


d  ( Uii\ 

dt  \  i'*  / 

s  = 


A  further  application  is  one  that  Klein j  has  given.  The 
principle  of  correspondence  in  algebraic  geometry,  due  to  Charles- 
Cayley-Brill,  was  proven  under  suitable  restrictions  by  Hurwitz  J 
with  the  aid  of  the  theta  functions  of  several  variables.  Klein 
constructs  a  proof  by  means  of  his  prime  function  along  the 
lines  of  Hurwitz's  proof,  and  the  present  prime  function  Q{t,  r) 
yields  the  same  results.  Both  in  Hurwitz's  and  in  Klein's  proofs 
details  are  omitted  which  can  be  satisfactorily  supplied  by  the 
theorems  above  considered  in  II,  §  6. 


*  Math.  Ann.,  36  (1889),  p.  23.  This  particular  canonical  surface  is  obtained 
from  Noether's  normal  curve  and  was  given  by  Klein  in  his  lectures  on  Abelian 
Functions,  Gottingen,  W.-S.,  1888/89. 

t  Gottingen  lectures  on  AbeUan  Functions,  W.-S.,  1888/89,  Lecture  IX, 
Dec.  15,  1888. 

X  Math.  Ann.,  28  (1887),  p.  561. 


functions  of  several  complex  variables.  225 

§  12.    The  Algebraic  Functions 

A  single-valued  function  on  F  having  no  other  singularities 
than  poles  yields  an  absolute  invariant  of  the  group  G,  the  new 
function,  considered  in  K,  being  analytic  except  for  poles. 

The  necessary  and  sufficient  condition  which  2?i  points  of  %, 
(Tk  and  Tk,  k  =  1,  ■  ■  • ,  fh  must  satisfy  if  they  are  to  be  the  zeros 
and  poles  of  such  a  function  F{t),  is  found  by  contour  inte- 
gration* to  be  the  following: 

(i)        «'„((7i)H \-u\{a„)  =  2i\iTi)-{ \-^l'a{Tn)   (mod. periods), 

i.  e., 

n  p 

(iO  S  K'"-    =  Ma  Tri  +     J2  Vj  (laj, 

where  a  =  1,  •  •  •,  p  and  the  n^,  Vj  are  integers. 

If  one  of  the  above  points,  as  r„,  be  replaced  by  an  equivalent 
point  under  the  group  G,  the  coefficients  ju^,  Vj  will  thereby  be 
modified.  It  is  evidently  possible  to  choose  the  latter  point  so 
that  these  coefiicients  will  all  be  zero,  and  we  suppose  this  done.f 

n 

(i'O  Z<^^^  =  0, 

where,  now,  ai,  •  •  - ,  <Jn  and  n,  •  •  ■,  r„_i  lie  in  %,  while  t„  may 
not. 

More  generally,  the  points  ci,  •  •  • ,  (Xn  and  xi,  ■  •  • ,  Tn  may  be 
any  2n  points  satisfying  (i")  and  such  that  the  points  of  % 
equivalent  under  G  to  o-i,  •  •  • ,  (r„  are  all  distinct  from  the  points 
of  %  equivalent  to  n,  •  •  •,  r„. 

To  express  the  function  F{t),  or  to  prove  the  existence  of  such 
a  function,  the  procedure  is  similar  to  that  in  the  elliptic  case, 
where  (T{t  —  r)  corresponds  to  the  present  Q.{t,  r).     Form  the 

*  The  integrals  which  are  to  be  extended  round  the  boundary  of  a  are 

^fd  log  F{t),     -i-.  r  wa{t)d  log  F{1). 

t  It  is  sufficient  for  the  application  which  follows  that  merely  the  Vj  be 
reduced  to  zero. 


226  THE  MADISON   COLLOQUIUM. 

function 

^  fig  (Tl)    '••   9.{t,  <Tn) 
^^^^  12(i,  Ti)    •  .  .    Q(t,  Tn)  ' 

This  function  has  the  desired  zeros  and  poles.     Moreover, 

n 

where  a  =  1,  •  •  •,  p,  and  this  completes  the  proof. 

§  13.    Parametric  Representation  of  a  Homogeneous 

Algebraic  Configuration 
Let 

(1)  Fiw,  z)  =  0 

be  an  irreducible  algebraic  equation.  Let  homogeneous  variables 
be  introduced.  If,  for  example,  we  wish  to  regard  (1)  as  a  curve 
in  the  projective  plane,  we  shall  set 

U) 

On  the  other  hand,  we  may  regard  (1)  as  a  curve  in  the  plane 
of  analysis,  and  set* 


X2 

X3 

z  =  —, 

IV  =  — 

Xi' 

Xi 

Z2 

W2 

2  —        > 

W  — 

Zl 

Wl 

Let  (1)  be  uniformized  as  above  by  the  function 

z  =  <p{t). 

Then  we  can  express  z,  iv  in  terms  of  t  by  the  aid  of  the  prime 
function  as  follows: 

■^      ^       n{t,  ai)  . . .  U{i,  an) '         "'       m,  «i)  •  •  ••  ^(t,  «..) ' 

Here,  the  points  of  (^  equivalent  to  Oi,  •  •  ■,  an  are  distinct  from 
those  of  5  equivalent  to  6i,  •  •  •,  6„;  and  similarly  for  ai,  •  •  -,  am 
and/3i,  ■",l3m. 
*Cf.  Riemann,  Abelsche  Funktionen,  §6;  Werke,  1.  ed.,  p.  103. 


FUNCTIONS   OF   SEVEILIL   COMPLEX   VARIABLES.  227 

To  uiiiformize  the  homogeneous  configuration  corresponding 
to  (B)  it  is  sufficient  to  set 


Zi  =   pQ{t,  Oi)    • 

■  •  9{t,  an), 

z,  =  p9.(t,h)  ■■ 

•  9{t,  6„), 

U'l  =  a  9.(f,  cvi)  • 

••  9(t,a„d, 

■W2  =  aQ{t,^i)  ■ 

■  •  P.{f,  13.,). 

A  fundamental  domain  is  given  by  the  cyHndrical  region 
{R,  S,  5)  corresponding  to  the  points  (p,  a,  t),  where  0  <  ]  p  [  <  oo , 
0  <  I  <r  I  <  CO ,  and  t  lies  in  %.  Aside,  then,  from  the  fact  that  %, 
regarded  as  a  fundamental  domain  for  the  automorphic  group  G, 
is  of  deficiency  p  >  1,  there  is  the  additional  circumstance  that 
both  R  and  S  are  doubly  connected  regions. 

The  number  of  independent  variables  of  the  allied  homo- 
geneous configuration  is  3. 

To  uniformize  the  configuration  which  corresponds  to  {A)  let 

^{t,  Ci)  •  •  •  ^{t,  ci)  ^. 

be  the  least  common  multiple  of  the  denominators  of  z  and  w 
in  (2).     Then  we  set 

.ri  =  p£l{t,  ci)  '•  •  ^{t,  ci), 

and  X2,  Xz  are  given  by  multiplying  p  by  a  suitable  product  of  fl- 
factors. 

A  fundamental  domain  is  here  furnished  by  the  cylindrical  region 
(R,  %),  corresponding  to  the  points  (p,  /),  where  0  <  |  p  |  <  qo  and 
t  lies  in  g. 

In  case  the  product  that  represents  Xi  contains  factors  not 
appearing  in  the  product  for  Zi,  the  number-pairs  (zi,  Zo)  will 
not  stand  in  a  one-to-one  and  continuous  relation  to  the  number- 
pairs  (a:i,  .To),  since  the  latter  will  contain  the  number-pair 
(0, 0), — a  fact  of  importance  in  the  theory  of  homogeneous 
variables. 

Such  uniformizations  as  the  foregoing  are  of  use  in  studying 


228  THE  MADISON   COLLOQUIUM. 

the  configurations  in  the  space  of  the  homogeneous  variables 
which  are  alHed  to  a  given  algebraic  configuration. 

§  14.      LiXEAK   DiFFEKENTIAL   EQUATIONS    ON   AN   ALGEBRAIC 

Configuration,  and  the  Factor  (p'{t) 

In  his  further  development  of  Riemann's  programme  relating 
to  the  determination  of  linear  differential  equations  by  their 
monodromic  group  Klein  has  studied  differential  equations  of 
the  type 

(A)  ^  +  P{u,  z)^-\-  Q{u,  z)U=0, 

where  the  coefficients  P  and  Q  are  single-valued  functions  on  a 
given  algebraic  Riemann's  surface  F  corresponding  to  the  irre- 
ducible algebraic  equation /(?f,  2)  =  0  of  arbitrary  deficiency  p; 
the  coefficients  having  no  other  singularities  on  F  than  poles,  and 
being,  therefore,  rational  in  ii,  z.  Let  P  and  Q  be  further  so 
restricted  that  the  singular  points  of  (A)  are  all  regular,  and  let 
p  >  1.  Among  such  differential  equations  the  subclass  is  of 
especial  interest  whose  members  have  no  singular  points  what- 
ever. If  Ui  and  U-z  be  two  linearly  independent  solutions  of 
such  an  equation  and  we  set* 

(1)  g=. 

then  the  neighborhood  of  an  arbitrary  point  of  F  is  mapped  by 
this  function  on  the  smooth  neighborhood  of  a  corresponding 
point  of  the  extended  s-plane  (or  sphere) . 

The  function  s  is  multiple-valued  on  F.  When  z  describes  a 
closed  path  on  F,  a  given  determination  of  s,  continued  analyti- 
cally along  this  path,  goes  over  into  a  linear  function  of  the  initial 
determination : 


*  The  notation  s  was  used  by  Schwarz  in  similar  cases,  to  whom  are  due  the 
oarliest  investigations  in  this  field  which  appeared  after  Riemann's  fundamental 
memoirs.     Klein  uses  the  letter  ij  in  this  sense. 


FUNCTIONS   OF   SEVERAL   COMPLEX  VARIABLES.  229 

ys  -{-  0 

Thus  s  is  homomorphic,  §  5. 

The  present  class  of  equations  contains  dp  —  3  parameters, 
and  it  is  a  problem  conceived  in  the  spirit  of  Riemann's  theory 
to  find  a  vital  property  of  the  solutions  which  shall  suflBce  com- 
pletely to  determine  these  parameters  and  thus  single  out  from 
the  class  a  unique  member. 

Klein*  has  given  the  following  solution  of  this  problem. 

Let  F  be  rendered  simply  connected  by  a  system  of  cuts,  the 
bounded  surface  being  denoted  by  F'.  Then  the  function  (1) 
will  map  F'  on  a  simply  connected  region  of  the  5-plane.  This 
region  will  have  no  branch  points;  but  it  may  overlap  itself,  and 
even  if  this  were  not  the  case,  the  further  regions  obtained  by 
allowing  z  to  cross  the  boundary  of  F'  and  then  describe  F'  again 
may  conceivably  overlap  one  another.  Let  the  totality  of  such 
regions  be  denoted  by  S. 

As  a  first  restriction  on  the  present  class  of  differential  equations 
Klein  demands  that  2  shall  be  simple,  i.  e.,  consist  of  a  single- 
sheeted  region. 

To  state  the  requirement  in  another  form,  it  is  this.  The 
function  s  is  multiple-valued  on  the  closed  surface  F.  And  now 
we  demand  that  the  values  which  s  takes  on  in  a  given  point  of 
F  shall  all  be  distinct,  no  matter  where  this  point  be  chosen. 

We  arrive,  then,  at  a  class  of  differential  equations  among 
those  under  consideration  whose  allied  function  s  is  such  that, 
by  means  of  it,  the  algebraic  configuration  f(u,  z)  =  0  can  be 
uniformized.  The  functions  of  s  that  here  present  themselves, 
namely  u  and  z,  are  single-valued  automorphic  functions.  But 
the  differential  equation  (A)  is  still  not  uniquely  determined. 

The  final  requirement  is  this.  The  region  2  shall  consist  of  the 
interior  of  a  circle.     It  is  still  possible  to  pass  from  one  circle  in 

*Math.  Ann.,  19  (1882),  p.  565;  20  (1882),  p.  49;  21  (1883),  p.  141.  Cf. 
also  Illein's  Gottingen  lectures,  Ueber  linearo  Differentialgleichungen  zweiter 
Ordnung,  1894  (lithographed). 


230  THE  MADISON   COLLOQUIUM. 

the  5-plane  to  any  other  by  a  linear  transformation  of  .9.  If, 
however,  we  regard  all  such  differential  equations  (A)  as  equiva- 
lent,— their  Schwarzian  resolvent 

[s]g=  R{u,  z) 

will  in  fact  be  the  same  for  all, — we  have  the  result  that  the 
differential  equation  {A)  is  uniquely  determined  by  the  above 
requirements.* 

Thus  it  appears  that,  when  an  algebraic  function  of  deficiency 
2?  >  1  is  given,  a  differential  equation  corresponding  to  it  can  be 
so  chosen  that  s  is  precisely  the  function  which  we  obtained  by 
conformal  mapping  as  t,  namely  (13)  in  §  5. 

From  the  foregoing  developments  we  conclude  that  the 
Abelian  integrals  corresponding  to  F,  when  considered  in  their 
dependence  on  their  parameters,  form  a  class  of  functions  which, 
in  important  respects,  is  incomplete.  The  factor  (p'(t)  =  ll{dildz) 
is  an  essential  accessory,  and  this  constituent  is  supplied  by  the 
linear  differential  equation  (A),  whose  parameters  are  determined 
in  the  spirit  of  Riemann  and  from  a  point  of  view  similar  to  that 
which  has  dominated  a  long  line  of  important  researches  in 
another  branch  of  modern  analysis, — I  refer  to  the  theorems  of 
oscillation  of  Sturm  and  Klein. 


*  The  demand  that  2  be  a  circle  is  not  the  only  one  which  leads  to  famiUar 
functions.  Thus  we  might  have  demanded  that  the  boundary  of  2  consist  of 
a  discrete  set  of  points, — discrete,  as  this  term  is  defined  in  the  author's  paper 
in  the  Annals  of  Math.  (2),  14  (1913),  p.  143.  We  should  then  have  been  led 
to  the  automorphic  functions  of  the  Schottky  type.  Again,  the  differential 
equation  (A)  would  have  been  uniquely  determined. 


c 


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